Small scale radial inflow turbine performance and pre-design maps for Organic Rankine Cycles

Accepted Manuscript Small Scale Radial Inflow Turbine Performance and Pre-Design Maps for Organic Rankine Cycles

Violette Mounier, Luis Eric Olmedo, Jürg Schiffmann PII:

S0360-5442(17)31851-0

DOI:

10.1016/j.energy.2017.11.002

Reference:

EGY 11790

To appear in:

Energy

Received Date:

12 July 2017

Revised Date:

19 September 2017

Accepted Date:

01 November 2017

Please cite this article as: Violette Mounier, Luis Eric Olmedo, Jürg Schiffmann, Small Scale Radial Inflow Turbine Performance and Pre-Design Maps for Organic Rankine Cycles, Energy (2017), doi: 10.1016/j.energy.2017.11.002

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ACCEPTED MANUSCRIPT Highlights



Updated Ns-Ds performance maps for designing small-scale ORC radial turbines



Analysis of the underlying phenomena delimiting the performance map boundaries



Sensitivity analysis on the radial turbine geometrical dependencies



Ns-Ds and experimentally validated 1D model predictions within an error band of ±4%

ACCEPTED MANUSCRIPT 1 2 3 4 5 6 7 8 9

SMALL SCALE RADIAL INFLOW TURBINE PERFORMANCE AND PRE-DESIGN MAPS FOR ORGANIC RANKINE CYCLES Violette MOUNIER (*), Luis Eric OLMEDO, Jürg SCHIFFMANN Laboratory for Applied Mechanical Design, Ecole Polytechnique Fédérale de Lausanne, EPFL Rue de la Maladière 71b, Neuchâtel, 2000, Switzerland (*) corresponding author:

10 11

Keywords:

12

Abstracts

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While small scale ORCs are currently dominated by volumetric expanders, the use of turbomachines is reconsidered due to their high efficiency and power density. Yet, suitable performance maps, which ensure an accurate starting point for the turbine design, are still missing for small-scale turbines. This paper proposes a new non-dimensional performance map tailored for small-scale turbines. The map is generated using an experimentally validated 1D code and adapted to small-scale ORCs applications. A new polynomial fit is proposed, which accounts for the pressure ratio, since it is suggested to have a strong influence on the shape of the map. Through the analysis of the turbine losses, the underlying phenomena shaping the efficiency map are explained. A sensitivity analysis of the geometrical dependencies on the map shows a strong impact of the shroud to tip radius ratio, and explains why the Ns-Ds surface of the presented map is smaller than the original one. Compared to the experimentally validated 1D model the new map yields prediction errors below 4%.

24

Nomenclature

Organic Rankine Cycle, small scale, radial turbine design, radial turbine efficiency, performance map, Ns-Ds maps

a

speed of sound

m.s-1

A

Area

m2

b

blade height

m

B

Blockage coefficient

-

C cnoz

absolute velocity

m.s-1

nozzle chord length Diameter specific diameter Clearance mechanical power Enthalpy loss coefficients Length Mass mass flow Mach number rotational speed rotor-speed-diameter product specific speed nozzle blade opening Pressure

m m m W kJ

D Ds e E h K L m m Ma N NDm Ns onoz P

m kg kg.s-1 rpm rpm.mm m Pa

ACCEPTED MANUSCRIPT Q r Re snoz T U V V W Z

heat flow Radius Reynolds number nozzle blade spacing Temperature tip speed velocity Volume volumetric flow relative velocity number of blades

W m m K m.s-1 m3 m3 s-1 m.s-1 -

25 26

27 28

Greek letters α β Δ ϵ 𝜁 ξ η υ ρ ω

nozzle angle rotor angle difference of/drop of Shroud to tip ratio blade height to tip ratio nozzle/interspace loss ratio Efficiency air gap sheer stress Density rotational speed

Subscripts abs back choke cl cond disc gen h hous i in inc is m noz o pass r rel rms rot s shaft sleeve splits stat-aero turb

Absolute rotor back face Choke Clearance Condensation axial disc Generator Hub Housing inner Inlet Incidence Isentropic Meridional Nozzle Outer Passage Radial Relative root mean square Rotor Shroud turbine shaft bearing sleeve Splitters static aerodynamic parts Turbine

° ° N.m kg.m-3 rad.s-1

ACCEPTED MANUSCRIPT

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throat thrust tot ts tt turb x

nozzle throat axial thrust Total total-to-static total-to-total Turbine Axial

Acronyms CR LR ORC PR RMSE SBA SP VR

Clearance Ratio Loss Ratio Organic Rankine Cycle Pressure Ratio Root Mean Square Error Shaft Bearings Assembly Size Parameter Volume Expansion Ratio

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1. Introduction

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Within the frame of decentralized energy conversion and savings, the significant potential of micro power generation through Organic Rankine Cycles (ORCs), which enable to recover energy from lowgrade heat, is now well-recognized. Applications of ORCs range from domestic co/tri generation power systems to energy recovery in the industrial and transportation sectors. While ORCs are well-known thermodynamic cycles, the selection and design of the ideal expander technology, which is one of the key components of the cycle, is still debated, in particular when small capacities are targeted. Literature for small scale ORC is dominated by volumetric machines, in particular by scroll and reciprocating machines [1–4]. Their main advantages are low specific cost, high robustness and reliability. Their disadvantage is that they usually require oil for lubrication of moving parts and bearings. Oil migrating within the cycle is known to depreciate heat transfer because of the differences in thermodynamic properties between oil and working fluid in terms of evaporation temperature and viscosity. This implies decreasing heat transfer coefficients [5–7] and increasing pressure losses [8], leading to the depreciation of the cycle thermal efficiency.

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The use of dynamic turbomachines has been identified as a promising way of increasing thermal efficiency since their operating principles do not involve parts sliding against each other, except for bearings, where contactless technologies such as magnetic or gas lubricated bearings can be used. Dynamic machines offer higher theoretical efficiencies and higher power density compared to volumetric machines, which are limited by their volume ratio capacity. For low power applications, scaling laws suggest that dynamic machines tend to be small requiring rotor high speeds, which leads to mechanical design challenges [9]. However, recent progress in gas lubricated bearing technology and in small scale turbomachinery has redrawn interest in this technology, in particular for ORC applications [10–14].

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Various authors have contributed to the optimization of ORC cycles, mainly by investigating optimum working fluids towards a specific application [15–19]. Unfortunately, the design of the turboexpander is often not considered thoroughly, and a fixed isentropic efficiency is used instead, regardless of the operating conditions. Nonetheless, as shown recently by Mounier et al. [20], designing small-scale ORC systems by coupling the thermodynamic and working fluid optimization with the turbo-expander preliminary design characteristics offers valuable insights into the trade-off mechanisms. Indeed, in addition to finding optimal thermodynamic layouts, the integrated approach allows to gain insights into the expander design.

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The classical design of turboexpanders undergoes four steps, each of them increasing the geometrical detail of the turbine stage. The initial 0D analysis yields the overall baseline of the turbine with a first guess of tip diameter and rotor speed. The second step leads to the 1D analysis, or ”mean line analysis”, which allows to add geometrical information to the inlet and the exhaust of the turbine design. The third step involves through flow calculations, often based on streamline curvature codes, which allows defining and tuning the complete rotor blade geometry. Finally, the design procedure ends with the 3D flow analysis, which allows analyzing and optimizing detailed flow patterns. The 3D-CFD approach is the most accurate but also the most expensive in computational time, which makes it unpractical for integrated optimization procedures. The 1D analysis is based on velocity triangles at the rotor inlet and exhaust and the efficiency is computed through enthalpy loss correlations. This approach is considerably less time consuming, and can be easily integrated into more complex design tools that require many iterations, such as multi-objective optimizations, thermodynamic system integration or dynamic analyses. Various authors have presented 1D models for radial inflow turbines [21–25]. Fiaschi et al [26,27] recently proposed a complete design procedure from the 0D to the 3D step, suggesting total to static isentropic efficiency deviations between the 0D and 3D approaches within 1.5%. Similar validation towards CFD has been achieved by Sauret et al. [28], Rahbar et al. [29] and Demierre [30]. In addition to a comparison between design tools Demierre et al. [11] offer an assessment of experimental data with regards to 1D-model predictions, suggesting a 5% difference on a wide range of rotor speeds and pressure ratios. This clearly shows that reduced order models can be used for coupled ORC system and turboexpander design, without incurring significant errors.

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However, in order to start the 1D-design with a good first guess, accurate non-dimensional efficiency and performance maps are needed. Such non-dimensional performance maps are often used in the 0D pre-design step and stem from the extensive work of Balje [31]. These maps are based on the principles of dimensional analysis and similarity concepts, and are widely used for pre-design purposes since they offer a convenient and practical approach to predict sizing, rotor speed and performance when operating at nominal conditions. However, these maps are challenged when applied to small-scale turbomachinery, since they are based on large-scale machines. Capata et al. [32] proposed a re-scaling of the original Balje maps for small-scale radial turbomachinery (Re≅105) by applying a Stodola correction factor fitted by means of CFD simulations. The numerical results suggest a 14.4 to 10-point efficiency drop compared to the original Balje maps. Da Lio et al. [33] proposed tailored performance maps for single stage axial turbines, based on the most recent loss correlations and tested with organic working fluids. They observed that the volume expansion ratio (VR) and the size parameter (SP) as defined by Macchi et al. [34] have a significant influence on the turbine efficiency. Hence, they proposed an adaptation of the Balje maps by taking the effect of these design parameters into account. Further, they have shown that while the VR-SP performance maps of an axial turbine follow the same qualitative trend for different working fluids, they present discrepancies in the efficiency estimation in the order of 1.5%. Since the Balje maps were obtained for non-organic media (air, combustion gases,…) they are often challenged by the ORC community [13].

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Nature of the Issue. It is well known that accurate pre-design phase data is essential for a time efficient turbomachinery design process. Unfortunately, the currently available turboexpander design maps by Balje are outdated, in particular in view of small-scale applications since downscaling phenomena such as low-Reynolds number and increased relative tip clearance effects are not considered [13,33]. Moreover, the original maps were built by using non-organic working fluids (air and fuel gas mainly). As a consequence, the lack of reliable pre-design performance and design maps for small-scale turbomachinery makes the pre-design phase challenging, due to the risk of obtaining a misleading geometry early in the design process. Moreover, the underlying phenomena defining the shape of the iso-efficiency contours in the specific speed and diameter plot are scarcely addressed in the literature.

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Goal and Objectives. The goal of this paper is to propose a new and updated non-dimensional design map for reduced scale radial inflow turbines and to analyze its features and limitations. The objectives are: (1) to update the current non-dimensional maps for radial inflow turbines based on experimentally

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validated 1D models to include downscaling effects, (2) to highlight the limitations of this nondimensional map based on the loss analysis at different pressure ratios and (3) to check the validity of the obtained performance maps.

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Scope of the paper. The methodology for the new performance map generation is presented initially, including details about the 1D model, the design variables and the map generation procedure. In a second step, the different performance maps are analyzed, especially by presenting the underlying phenomena that determine the boundaries of the obtained performance maps. Further, the new efficiency map is compared with the original Balje map. The validity of the new efficiency map is checked, by directly comparing its results with the experimentally validated 1D model.

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2. Generation of the performance map

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1D radial inflow turbine models are essentially based on velocity triangles at the rotor inlet and exhaust. For the reduced order model used in this analysis the aerodynamic performance is corrected by loss correlations introduced by Baines [35]. These correlations account for (1) nozzle and nozzle interspace losses, (2) incidence, passage, tip clearance and trailing edge losses on the rotor, and (3) diffusion losses. More details about the 1D radial inflow turbine code used for this work can be found in [11,30]. The 1D model has been validated both with CFD [30], and with experimental data, obtained with a 2 kW 18 mm tip diameter radial inflow turbine used in an ORC driven heat-pump system running on R134a [11]. The predicted turbine isentropic efficiency and the measured data yield a relative error of ±5% for pressure ratios below 4.2 and for rotor speeds ranging from 150 to 200 krpm suggesting a very good agreement between predicted and experimental data. Figure 1 shows the radial inflow turbine layout, main components and basic geometries.

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The main advantage of 1D models is that while being relatively simple, off-design performance is well captured. However, when combined with complex optimization of thermodynamic systems, the efficiency rating of the turbine through the 1D analysis remains time consuming, particularly when only an estimation of the turboexpander overall features, namely tip diameter, rotational speed and efficiency, is required. For that purpose, using an efficiency chart that allows for a quick evaluation would be of great interest for accurate system pre-design.

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In order to obtain accurate and unified performance maps for turboexpanders in the range of 1-10 kW power, that cover a wide range of both operating conditions and geometries, a Monte Carlo simulation approach was selected by applying it to the experimentally validated 1D turbine. In a second step, the obtained data cloud allowed to tailor a new 0D performance map, or Ns-Ds performance map, based on non-dimensional specific speed Ns and diameter Ds for reduced scale radial inflow turbines. The definition of the specific speed Ns and diameter Ds is given in equation 1, where D is the turbine rotor tip diameter, V the volumetric flowrate at the turbine exhaust, Δhis the isentropic enthalpy difference along the turbine, and ω the rotational speed.

Ns = |

ω V0.5 Δhis|0.75

Ds =

D |Δhis|0.25 V0.5

(1)

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The Monte Carlo simulation was performed by using 65’000 randomly and equally distributed geometrical and operational configurations (Table 1). The turbine tip diameter and rotor speed ranges have been set for a shaft mechanical power range of [1-10] kW based on the existing Balje maps. The turbine inlet temperature is set to typical micro-scale ORC systems ranges. The turbine outlet pressure is calculated from ORC condensation temperatures ranging from 20°C to 60°C, while a variable pressure ratio from 1.5 to 8.5 sets the turbine inlet pressure.

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The geometric design parameter dependencies (diameters, angle, blade height,…) were selected based on design guidelines given by Balje [31] and Baines [35]. Finally, it has been assumed that the turbine operates at the onset of choking conditions in the nozzle throat, which enables to maximize the turbine stage work output. The nozzle outlet angle α3 is therefore adapted such that the choked mass flow at the nozzle throat is the same as the specified mass flow mturb, as proposed in equations 2 and 3, neglecting the effect of the nozzle blade trailing edge thickness. Hence, this methodology enables to set the required nozzle outlet angle such as to avoid a mismatch between the nozzle and the rotor. With the use of advanced nozzle designs with variable throat areas [36], a radial turbine could adapt to different mass flows, which is of particular interest at part load. The nozzle outlet angle is given as follows: α3 = cos ‒ 1

(

Achoke =

Achoke 2πr3b3(1 ‒ Bthroat,noz) mturb ρthroat,nozCthroat,noz

)

(2) (3)

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where ρthroat,noz and Cthroat,noz are the fluid density and absolute velocity and the nozzle throat. Bthroat,noz represents the aerodynamic blockage coefficient of the nozzle, ranging between 0 and 10 % [35]. Bthroat,noz = 7% has been selected, since it is well fitting the experimental data and it corresponds to best engineering practice.

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The evaluated solutions are subject to the following constraints: 

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Only feasible solutions are kept for generating the map and a 5th order polynomial function is applied to draw the isentropic efficiency lines within the Ns–Ds diagram. Besides varying pressure ratios, the Monte Carlo simulation has also been performed for fixed pressure ratios of 2, 4, 6 and 8 and for four typical organic working fluids (R134a, R245fa, R152a and R600a).

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3. Results and discussions

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3.1

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Figure 2 represents the 80% isentropic efficiency contours in the Ns-Ds map obtained for fixed pressure ratios across the turbine and for R134a. The surface within the iso-efficiency contours is suggested to decrease while a shift towards lower specific diameters Ds and higher specific speeds Ns is observed with increasing pressure ratios. In addition, while the maximum Ns and minimum Ds seem unaffected by the pressure ratio, the maximum specific diameter decreases and the minimum specific speed increases with pressure ratio.

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The influence of the pressure ratio on the 0D performance map affects its accuracy when different pressure ratios are used for its generation. Indeed, when using the polynomial fit at varying pressure ratio (black line on Figure 2), the Ns and Ds values are averaged, meaning that the isentropic efficiency might be underestimated at low-pressure ratios, and vice versa at high-pressure ratios. The influence of the pressure ratio is caused by the evolution of the share of the different losses in the turbine stage. In order to capture this effect, the authors propose to include the turbine pressure ratio as an input variable to the 0D performance map. Using a multivariate 5th-order polynomial fitting, a new tailored equation



The vapor quality must be above unity at each station, in order to avoid flashing and premature blade erosion and bearings failure as a consequence The turbine geometry needs to achieve the specified pressure ratio within a deviation 0.5%

Influence of the pressure ratio

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rating the total-to-total isentropic efficiency is built as a function of Ns, Ds and pressure ratio, presenting an R2 in excess of 0.98 and a RMSE of 0.026. The polynomial fit can be found in [37].

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Figure 3 shows similar efficiency contours obtained for the investigated fluids as a function of pressure ratio across the turbine. The maximum RMSE between the polynomial fits built for each working fluid is equal to 0.012, which is below the accuracy of the map (RMSE=0.026). This results in maximum average absolute errors below 1-pt in isentropic efficiency from one fluid to the other across the map, which corroborates observations by Da Lio [38]. Since these performance maps are used for pre-design purpose, the results suggest that the proposed map based on R134a is reasonably well suited to a wide variety of ORC working fluids (critical temperatures ranging from 101 to 154°C).

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In order to further simplify the turbine design it is common practice to approximate the ideal radial inflow turbine design variables along the “efficiency ridge”, which is often approximated by the NsDs = 2 curve [35]. This correlation, similar to the Cordier line for compressors, is represented in Figure 4 for different pressure ratios. Although this correlation fits well the original Balje diagrams, it clearly leads to suboptimal designs in the new maps, in particular for operation off the maximum efficiency. As a consequence, a new set of polynomial equations, estimating the isentropic efficiency (equation 4) and optimal specific diameters (equation 5) as a function of both pressure ratio and specific speed is introduced as follows: (4 )

ηis = 0.45 + 1.64Ns ‒ 0.05PR ‒ 2Ns2 + 0.15NsPR + 0.63Ns3 ‒ 0.07Ns2P ‒ 0.003NsPR2 Ds

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(5 2PR = 9.42 ‒ 18.3Ns ‒ 0.16PR + 17.8Ns2 + 0.24NsPR + 0.001PR2 ‒ 6.37Ns3 ‒ 0.13Ns ) + 0.001NsPR2 The resulting curves along the new “efficiency ridges” in the Ns-Ds maps are also plotted in Figure 4 highlighting the difference with the classical NsDs=2 curve. The selection of the specific speed optimizing the turbine efficiency (equation 4) leads immediately to the appropriate specific diameter, and hence the turbine diameter. However, this method might depreciate the overall system design, since the turbine diameter might need to be adjusted when additional constraints such as volume, mass or rotordynamic stability have to be addressed.

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3.2

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In order to investigate and highlight the underlying phenomena giving the iso-efficiency contours their characteristic shape in the Ns-Ds-diagram the different loss mechanisms are analyzed in detail. Note that the losses are normalized with the isentropic enthalpy drop across the nozzle or the rotor depending on their nature (stationary nozzle or rotor).

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Nozzle and interspace passage losses. Figure 5 and Figure 6 represent the relative nozzle and interspace passage losses for various pressure ratios. The losses are normalized by the isentropic enthalpy drop through the nozzle and interspace passage respectively. The plots clearly suggest that the overall turbine efficiency drop towards high Ds values is a consequence of the loss increase in the nozzle and in the interspace passage. The loss ratio (LR), defined by equation 6 for the nozzle stage is a function of ξ (equation 8), which represents the nozzle/interspace loss coefficient defined by Rodgers [39] and proposed by Baines [40]. Detailed expressions for onoz, cnoz and snoz which correspond to the nozzle throat opening, nozzle blade chord length and nozzle blade spacing, can be found in [11], whereas the Re-number is based on the flow velocity and blade height at the nozzle throat.

Influence of the turbine loss mechanisms on the efficiency contours

LR =

∆hnoz ∆his,noz

=

ξCthroat,noz2 2∆his,noz

=

ξ ξ+1

(6)

ACCEPTED MANUSCRIPT

Cthroat,noz2 = ξ=

Cis,throat,noz2

0.02

(

ξ+1 tan α32

Re0.05 snoz cnoz

=

+

2∆his,noz

(7)

ξ+1

)

onoz b3

(8)

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It appears that the nozzle losses are governed primarily by the geometric nozzle outlet angle, which is adapted in order to obtain a choked flow at the specified mass flow. With a diameter increase and/or a volume flow reduction, which both lead to higher Ds values, the nozzle outlet angle α3 grows, and the nozzles losses increase as a consequence. This effect is amplified with pressure ratios due to the higher density at the throat. For a given mass-flow, an increased pressure ratio decreases the required area (α3 increase) for achieving choked nozzles.

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The evolution of the relative interspace losses in the Ns-Ds diagram is represented for various pressure ratios in Figure 6. In a similar manner as for the nozzle losses, the interspace losses are mainly driven by the absolute flow angle (α4), which is essentially governed by α3. Hence, higher flow angles driven by increased diameter and reduced volume flows increase the interspace losses and the iso-loss lines follow the same trends as for the nozzle losses in Figure 5. For a pressure ratio of 2 the interspace losses are negligible and therefore no contours are visible.

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Rotor incidence losses. The rotor blade incidence losses are calculated based on a model proposed by Wasserbauer et al. [41]: ∆hinc =

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(9)

where β4,opt = -40° is the optimal flow angle at the rotor inlet [42] and W4 the relative flow speed defined by: W4 =

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1 2 (W sin (β4 ‒ β4,opt)2) 2 4

U24 + C24 ‒ 2U4C4sin β4

(10)

The evolution of the relative incidence losses in the Ns-Ds-diagram is represented in Figure 7. Note that in the rotor the losses are normalized by the isentropic enthalpy drop across the rotor. The incidence loss = W4 a ) ratio contours coincide with the evolution of the relative Mach number at the rotor inlet (M 4,rel

4

as represented in Figure 8, which suggests that these losses mainly depend on the rotor inlet Mach number. At high Ns and Ds values, and therefore high rotor speed and/or tip diameter, the rotor tip speed U4 = r4ω increases, which drives the incidence losses. However, at low Ns and Ds values, as the tip speed U4 gets lower, the absolute rotor inlet velocity C4 needs to be increased in order to match the specified pressure ratio. This corroborates with the evolution of the absolute rotor inlet Mach number ( M = C4 a ) presented in Figure 9. Although the absolute rotor inlet Mach number increases with 4,abs

4

pressure ratio, the latter has little impact on the incidence losses. Both tip speed and absolute velocity increase with the pressure ratio in order to match the required turbine specific work, however, their effect on W4 is compensated by the 2U4C4sin β4 term, which further increases with pressure ratio. Tip clearance losses. The tip clearance losses are predicted through the correlation given by Baines [35] and described in equations 11 and 12. ex and er are the axial and radial tip clearances. The discharge parameters values Kx = 0:4, Kr = 0:75 and Kxr =-0:3 are recommended based on experimental data [35]. As they are mainly driven by tip speed (U4) the tip clearance losses increase for high values of Ns and Ds. These losses increase with the imposed pressure ratio, as the tip speed U4 increases in order to match

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the required Euler specific work defined by equation 13. This results in narrower Ns-Ds contours when the required pressure ratio increases, as shown in Figure 10. ∆hcl =

U34(Zblades + 0.5.Zsplits)

8π (r ‒ e 1 ‒ 6s r) r

Gx =

4

Cm4b4

(KrerGr + KxexGx + Kxr Gr =

(

erexGxGr)

(11)

)

(12)

r6s ‒ er Lrot ‒ b4 r4

∆hrot = U4Cθ4 ‒ U6Cθ6

Cm6r6rms

(13)

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Rotor passage losses. Figure 11 depicts the evolution of the passage loss ratio, which tend to rise towards low Ds and high Ns values and towards both high Ds and Ns values. The correlation for the passage losses implemented in the 1D model is proposed by Futral et al. [43] Benson [44], and Wasserbauer et al. [41] as follows: 1 ∆hpass = Kpass(W24cos (β4 ‒ β4,opt)2 + W26) 2

(14)

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where KPass = 0.3 has been identified experimentally. The passage loss ratio is governed by the relative rotor outlet velocity (W6), which corroborates with the analysis of the relative Mach number at the = W6 a ) as highlighted in Figure 12. A decrease of the specific diameter Ds due turbine outlet (M 6,rel

6

to the drop of the tip diameter r4, reduces both the exhaust hub and shroud diameters r6s and r6h, and hence the rotor throat area, thus forcing the relative exhaust velocity to increase. On the other hand, an increase of both Ns and Ds result in higher tip speeds, which drive both the relative inlet and the exhaust velocities.

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In addition, W6 is suggested to rise with pressure ratio, which is an inherent consequence of the increased specific turbine work (equation 13). In fact, higher-pressure ratios lead to higher tip speeds U4, which inevitably increase U6. An analysis of the exhaust velocity triangle clearly shows that W6 needs to increase with U6 in order to minimize the exhaust swirl (Cθ6). Note that the passage loss ratio does not exceed 15% for overall isentropic turbine efficiencies above 80%, independently of the pressure ratio.

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If W6 is high enough to choke the rotor throat, the flow angle at the rotor throat is corrected to allow for a supersonic expansion. Yet, this expansion is restricted and Mrel,6 is in practice limited to 1.1, as suggested in Figure 12. The pressure ratio has a limited effect on the passage losses, since the cos (β4 ‒ β4,opt)2 term decreases with pressure ratio, thus counterbalancing the W6 rise. Yet, it has been observed that W6 is limited to Mach 1.1 since only a limited amount of over-expansion can be achieved in the rotor passage. This is due to the axial Mach number at the rotor outlet that cannot exceed unity.

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Trailing edge losses. Finally, the trailing edge losses, which are based on correlations by Baines [40] are presented in Figure 13. These losses are one order of magnitude lower compared to the other rotor and nozzle losses and have therefore a negligible effect on the efficiency contours in the Ns-Ds diagram.

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As summarized in Figure 14, analyzing each loss mechanism individually along the turbine flow has allowed to identify the key aerodynamic parameters and underlying loss mechanisms that influence the shape of the iso-efficiency contours in the Ns-Ds diagrams as follows:

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1. The nozzle outlet angle α3 limits the map towards high values of Ds due to its influence on the nozzle losses. 2. The tip speed U4 mainly drives tip clearance losses and increases the rotor losses at high specific diameters and speeds. 3. The absolute velocity C4 at the nozzle exit drives the incidence losses and therefore constrains the width of the elliptic iso efficiency contours and elongates the contours along a constant NsDs product. 4. The relative velocity W6 at the rotor outlet limits the contours towards low Ds values due to increasing passage losses and Mach number at the rotor throat.

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3.3

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With the nozzle outlet angle being adapted for each operating condition, the turbine design is fully determined by the values of rotational speed and diameters obtained on the presented Ns-Ds map. The following section presents a sensitivity analysis showing the influence of geometrical design dependencies that have been applied as constant values to generate the map according to best engineering practices such as suggested by Baines [35]. They are the blade height to tip radius ratio, the relative clearance ratio and the shroud to tip radius ratio. Note that the plotted efficiency contours obtained by the sensitivity analysis correspond to the averaged pressure ratio iso-efficiency lines (black solid line of Figure 2).

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Figure 15 depicts the influence of the blade height to tip radius ratio ζ = b4 r4 on the 0.8 and 0.7 efficiency contours, when ranging from 10 to 30 %. At high ζ ratio, the nozzle outlet angle α3 increases due to the higher blade heights at the rotor tip. Higher nozzle angles, however, increase the nozzle losses, which decrease the specific diameters Ds in order to obtain the same isentropic efficiency. As a consequence, the results suggest that smaller blade height ratios allow to increase iso-efficiency surface in the Ns-Ds diagram since the nozzle flow angles tend to become smaller, which reduces the losses.

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Figure 16 represents the 0.8 and 0.7 efficiency contours obtained at clearance ratios CR = er,x b4, ranging from 2 to 30 %. The increase of the clearance ratio tends to shift the Ns-Ds map towards lower values of Ns and Ds, due to the influence of the clearance gaps in equation 11, which corroborates with the loss analysis above. Moreover, one can observe that the values of Ds must be smaller at high clearance ratios in order to ensure an isentropic efficiency of 0.8 at a given specific speed. This can be explained by the fact that the clearance losses are driven by both the tip speed and the tip clearance. Hence, at a larger tip clearance the tip speed needs to be reduced in order to keep the efficiency constant. Note that that these contours could increase even further without the clearance minimum set to 80 μm for manufacturing and mounting issues.

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Finally, Figure 17 represents the 0.8 and 0.7 efficiency contours obtained for shroud to tip radius ratios ϵ = r6s r ranging from 0.4 to 0.9. The results suggest that tuning the shroud to tip radius ratio allows

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Sensitivity analysis

4

to operate at higher specific speeds (increased ϵ) and higher values of specific diameters (decreased ϵ) which might offer interesting design options in cases where constraints due to operating conditions or manufacturing need to be taken into account. Note, however, that only ϵ values ranging from 0.6 to 0.8 allow to achieve efficiencies in excess of 80%. For shroud to tip radius ratio of 0.7 the surface for the 80% and 70% efficiency contours are larger than for the other values, which might offer potentially higher off-design performance. The corresponding polynomial fits are published in [37]. These plots also provide keys to understanding why, when compared to the original Balje maps, the new radial turbine isentropic efficiency contours delimit smaller Ns-Ds surfaces for a given total to static isentropic efficiency, such as shown in Figure 18:

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1. In order to compute the maps, Balje used varying shroud to tip radius ratios (ϵ = r6s r4) with values ranging from 0.4 to 0.8, as opposed to a fixed value of 0.7 used in this investigation as according to Baines [35].

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2. The original Balje maps were computed using a tip clearance ratio of 2 %. In the frame of this investigation, reduced scale rotors are considered, hence the minimum tip clearance is limited to 80 μm in order to stay within realistic manufacturing tolerances. In some cases, this threshold leads to tip clearance ratios approaching 20%, leading to similar results illustrated by the Figure 16. Furthermore, while Balje considered leakage losses as a part of the endwall friction losses [31], the correlation used for the present Ns-Ds map consists in a more fundamental analysis of the flow through the clearance gap, which has been experimentally validated [35].

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3. The Balje maps were computed for large-scale air machines, operating with significantly lower pressure ratios than the ones considered in this analysis, which tends to increase the efficiency contours in the Ns-Ds diagram, as suggested in Figure 2.

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3.4

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Compared to the 1D radial inflow turbine model, the Ns-Ds performance maps introduced in this investigation represent a significant reduction of the model order. In order to assess the effect of this simplification, the ratings obtained by the 0D and the 1D models are compared for a radial inflow turbine of 18 mm tip diameter that was also used to experimentally validate the 1D model. Different operating points within the conditions summarized in Table 2 have been tested, which correspond to typical values of a small capacity ORC without regeneration [45]. The turbine mass flow is directly computed through the evaporator power input, assuming that all the heat transfers are isobaric. Further assumptions used to calculate the cycle are steady state flow conditions and an isentropic pump efficiency of 60%.

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Two different calculations are implemented for the Ns-Ds map based evaluations, one which controls whether the optimum solution lies within the convex envelope generated by the original dataset used to build the new Ns-Ds maps (constrained), and another, which also retains solutions out of the convex envelope (unconstrained).

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Figure 19 compares both the optimum isentropic efficiencies and rotational speeds as a function of the pressure ratio resulting from both the Ns-Ds map and the 1D code for a constant ORC evaporator power of 22 kW. Both unconstrained and constrained solutions are displayed. As expected, the isentropic efficiency decreases with pressure ratio as a consequence of the increased nozzle and interspace losses. Similarly, the rotational speed increases with the pressure ratio to match the new turbine power requirements. Compared to the 1D-predictions both the optimum Ns-Ds rotational speeds and isentropic efficiencies are contained within a deviation band of ±2 and ±4.5% respectively.

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Figure 20 compares the optimum rotor speeds and isentropic efficiencies obtained by the 0D and the 1D models independently of the evaporation power. The 1D and 0D based predictions are contained within an error band of ±4 and ±7% for the rotor speed and the isentropic efficiency respectively. Finally, no significant difference occurs between the constrained and unconstrained solutions, meaning that the new Ns-Ds performance maps can be used on a wide range or operating conditions to predesign small-scale radial inflow turbines, without incurring significant prediction errors.

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3.5

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The new 0D performance map is generated through the evaluation of an experimentally validated 1D model fed with inputs summarized in Table 1, where most of the dimensions are scaled with the tip radius according to best design practice. From a sensitivity analysis, it appears that these geometrical dependencies are valid for small-scale turbomachinery as well. However, the analysis of the interactions between the different design variables is missing, and an integrated optimization of the 1D dimensions may lead to higher efficiencies.

Validation of the Ns-Ds map

Limitations of the maps and of the methodology

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Moreover, the present method imposes the turbine to operate at the onset of choke, a condition that might yield inaccurate prediction especially at low and high pressure ratios. Indeed, while this condition maximizes the work output by increasing the value of Cθ4 and avoids a mismatch between the nozzle and the rotor design, it may induce higher losses in the nozzle passage, in particular for low pressure ratios which could easily be achieved without a choked nozzle.

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Furthermore, the 1D enthalpy losses have been calculated with empirical loss correlations, which have not been extensively studied in the small-scale domain so far. More experimental data on a wider range of turbine geometries, fluids and operating conditions would be needed to further refine and validate the individual loss correlations at reduced scale. Nonetheless, the 1D model remains reliable since it fits the experimental data of a small scale turbine well on a wide range of operating conditions.

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Since the 1D-code has been validated on a given turbine geometry (see Table 1), the ± 4% error range on the performance map is strictly valid only for this optimal geometrical parameter set. It can be assumed that for varying geometrical ratios, in particular for the shroud to tip radius ratio, the deviation range could differ. In order to gain higher confidence levels on a wider geometrical range more experimental data is needed, which, unfortunately, is currently not available.

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Finally, it has to be noted that while these maps offer a powerful pre-design tool, they cannot be used for rating off-design operation.

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Conclusion

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An updated Ns, Ds performance map for micro radial inflow turbines based on an experimentally validated 1D model has been introduced. This new efficiency map has been generated with a set of thermodynamic and geometrical variables, using fixed geometrical dependencies found in the literature and from earlier prototypes. Since the effect of the pressure ratio is not to be neglected, a new model for predicting the total-to-total isentropic efficiency has been built, including the pressure ratio as an input along with the specific speed and diameters. The results suggest that the higher the pressure ratio the smaller the surface of an iso-efficiency contour in the Ns-Ds-diagram.

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An in-depth analysis of the losses occurring within the complete turbine stage has then been performed, highlighting the underlying aerodynamic phenomena shaping the performance map contours in the NsDs-maps. The nozzle and interspace losses set an upper limit for the values of Ds, due to the effect of the nozzle outlet angle. Since the nozzle outlet angle increases with increasing density at the nozzle throat it also explains why the performance map is highly dependent on the pressure ratio. When analyzing the losses in the rotor passage, it appears that the tip leakage losses limit the map towards high values of Ds and Ns, due to too high tip speeds resulting from high Ds and Ns values. On the other hand, the incidence losses are limiting the width of the elliptic shape of the Ns-Ds contours, due to the increase of the absolute velocity at the rotor inlet. Finally, the passage losses and choking conditions at the rotor throat limit the map towards lower values of Ds.

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Since this map was generated with fixed geometrical dependencies, a sensitivity analysis has been performed in order to investigate their effect on the map. It is suggested that both clearance and blade height to tip radius ratios have a significant influence on the available surface of the Ns-Ds map, concluding on the necessity to have them as small as possible, provided manufacturing feasibility. The ratio between shroud and tip radius also showed a strong influence on the map surface with a shift from high values of Ds to lower ones. This phenomenon is one of the reasons why the Ns-Ds surface of the presented map is much smaller than the original provided by Balje. Besides, Balje generated its maps with large-scale air machines and with out of date tip leakage loss correlations, emphasizing the difference even further.

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Finally, the new efficiency maps for radial inflow turbines have been validated with the direct use of the experimentally validated 1D code, showing a maximum error on the optimal isentropic efficiency map within 4% and hence providing an accurate and quick pre-design method for radial inflow turbines embedded in small-scale ORCs systems.

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ACCEPTED MANUSCRIPT Figure 1 : Schematic view and design variables of a typical radial inflow turbine with a volute, nozzle , rotor and a diffuser section. ......................................................................................................................2 Figure 2 : 80% iso-efficiency contours in the Ns-Ds diagram generated at fixed pressure ratios (colored lines) and for varying pressure ratio (black line)......................................................................................3 Figure 3 : 80% iso-efficiency contours in the Ns-Ds diagram for various pressure ratios and for typical ORC working fluids suggesting only limited influence of the working fluid on the efficiency contours. ..................................................................................................................................................................4 Figure 4 : Iso-efficiency contours in the Ns-Ds diagram generated at fixed pressure ratios superposed by the optimal correlation between Ns and Ds and by the theoretical curve NsDs=2. ............................5 Figure 5 : Nozzle loss ratios in the Ns-Ds diagram at different pressure ratios ......................................6 Figure 6 : Interspace loss ratios in the Ns-Ds diagram at different pressure ratio....................................7 Figure 7 : Incidence losses ratio in the Ns-Ds diagram at different pressure ratios .................................8 Figure 8 : Evolution of the relative Mach number at rotor inlet in the Ns-Ds diagram at different pressure ratios .........................................................................................................................................................9 Figure 9 : Evolution of the absolute Mach number at rotor inlet in the Ns-Ds diagram at different pressure ratios .......................................................................................................................................................10 Figure 10 : Clearance losses ratio in the Ns-Ds diagram at different pressure ratios.............................11 Figure 11 : Passage losses ratio in the Ns-Ds diagram at different pressure ratios ................................12 Figure 12 : Evolution of the relative Mach number at rotor outlet in the Ns-Ds diagram at different pressure ratios .........................................................................................................................................13 Figure 13 : Trailing edge losses ratio in the Ns-Ds diagram at different pressure ratios .......................14 Figure 14 : Underlying loss mechanisms limiting the boundaries of the iso-efficiency contours in the Ns-Ds diagram. .......................................................................................................................................15 Figure 15 : Ns-Ds contours when performed at different blade height to tip radii ratios for isentropic efficiencies of 0.8 and 0.7.......................................................................................................................15 Figure 16 : Ns-Ds contours when performed at different clearance ratios for isentropic efficiencies of 0.8 and 0.7 ..............................................................................................................................................16 Figure 17 : Ns-Ds contours when performed at different shroud to tip radii ratios for isentropic efficiencies of 0.8 and 0.7.......................................................................................................................16 Figure 18 : Comparison of the new performance map with the original map of Balje. .........................17 Figure 19 : Comparison of the optimum rotational speeds and efficiencies versus the pressure ratio, for the Ns-Ds based (constrained and unconstrained) and the 1D model for a fixed heat input of 22 kW..17 Figure 20 : Comparison of the optimum rotational speeds and efficiencies versus the pressure ratio, for the Ns-Ds based (constrained and unconstrained) and the 1D model, when the ORC heat input is varied ................................................................................................................................................................18

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Figure 1 : Schematic view and design variables of a typical radial inflow turbine with a volute, nozzle , rotor and a diffuser section.

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Figure 2 : 80% iso-efficiency contours in the Ns-Ds diagram generated at fixed pressure ratios (colored lines) and for varying pressure ratio (black line).

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Figure 3 : 80% iso-efficiency contours in the Ns-Ds diagram for various pressure ratios and for typical ORC working fluids suggesting only limited influence of the working fluid on the efficiency contours.

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Figure 4 : Iso-efficiency contours in the Ns-Ds diagram generated at fixed pressure ratios superposed by the optimal correlation between Ns and Ds and by the theoretical curve NsDs=2.

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Figure 5 : Nozzle loss ratios in the Ns-Ds diagram at different pressure ratios

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Figure 6 : Interspace loss ratios in the Ns-Ds diagram at different pressure ratio

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Figure 7 : Incidence losses ratio in the Ns-Ds diagram at different pressure ratios

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Figure 8 : Evolution of the relative Mach number at rotor inlet in the Ns-Ds diagram at different pressure ratios

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Figure 9 : Evolution of the absolute Mach number at rotor inlet in the Ns-Ds diagram at different pressure ratios

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Figure 10 : Clearance losses ratio in the Ns-Ds diagram at different pressure ratios

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Figure 11 : Passage losses ratio in the Ns-Ds diagram at different pressure ratios

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Figure 12 : Evolution of the relative Mach number at rotor outlet in the Ns-Ds diagram at different pressure ratios

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Figure 13 : Trailing edge losses ratio in the Ns-Ds diagram at different pressure ratios

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Figure 14 : Underlying loss mechanisms limiting the boundaries of the iso-efficiency contours in the Ns-Ds diagram.

Figure 15 : Ns-Ds contours when performed at different blade height to tip radii ratios for isentropic efficiencies of 0.8 and 0.7

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Figure 16 : Ns-Ds contours when performed at different clearance ratios for isentropic efficiencies of 0.8 and 0.7

Figure 17 : Ns-Ds contours when performed at different shroud to tip radii ratios for isentropic efficiencies of 0.8 and 0.7

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Figure 18 : Comparison of the new performance map with the original map of Balje.

Figure 19 : Comparison of the optimum rotational speeds and efficiencies versus the pressure ratio, for the Ns-Ds based (constrained and unconstrained) and the 1D model for a fixed heat input of 22 kW.

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Figure 20 : Comparison of the optimum rotational speeds and efficiencies versus the pressure ratio, for the Ns-Ds based (constrained and unconstrained) and the 1D model, when the ORC heat input is varied

ACCEPTED MANUSCRIPT Table 1 : 1D micro-turbine inputs for Ns, Ds maps generation based on the original Balje maps [37], typical ORC operating conditions, and geometric design guidelines recommended by Baines [36].......1 Table 2 : Boundary conditions for the comparison between the Ns-Ds performance map and the 1D radial inflow turbine model ......................................................................................................................1 Table 1 : 1D micro-turbine inputs for Ns, Ds maps generation based on the original Balje maps [37], typical ORC operating conditions, and geometric design guidelines recommended by Baines [36]. Term Maps variables Rotational Speed Rotor tip radius Turbine Inlet Temperature Condensation Temperature Pressure Ratio Mass flow Fixed parameters Rotor inlet angle Rotor outlet angle Rotor blades number Rotor splits blades number Nozzle blade number Dependent parameters Nozzle outlet blade height Nozzle inlet blade height Rotor blade height Nozzle outlet radius Nozzle inlet radius Rotor exit shroud radius Rotor exit hub radius Diffuser exit radius Rotor radial clearance Rotor axial clearance Rotor backface clearance Rotor Length

b3

Range

Unit

N r4 Tturb,in Tcond PR mturb

50-400 5-50 90-250 20-60 1.5-8.5 20-400

krpm [mm] [°C] [°C] [-] [g/s]

β4 β6 Zblades Zsplits Znoz

0 -60 9 9 5

° ° [-] [-] [-]

b1 b4 r3 r1 r6s r6h r7 er ex eback Lrot

0.11 ∙ r4 b3 b3 1.055 ∙ r4 2 ∙ r4 0.7 ∙ r4 0.3 ∙ r6s 1.1 ∙ r6s max (0.08,0.02 ∙ b4) max (0.08,0.02 ∙ b4) max (0.08,0.02 ∙ b4) 0.56 ∙ r4

[mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm]

Table 2 : Boundary conditions for the comparison between the Ns-Ds performance map and the 1D radial inflow turbine model Term ORC evaporator power Turbine inlet temperature ORC condensation temperature Turbine pressure ratio Turbine tip diameter Working fluid

Range [20-32] 160 35 [2-8] 18 R-134a

Unit kW °C °C [-] [mm]