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https://doi.org/10.1016/j.apenergy.2014.08.076

Three-Dimensional Off-Design Numerical Analysis ofan Organic Rankine Cycle Radial-Inflow Turbine

Emilie Sauret, Yuantong Gu∗

Queensland University of Technology, Brisbane, Australia

2 George Street, Brisbane, Qld 4000, Australia

Abstract

Optimisation of Organic Rankine Cycles (ORCs) for binary cycle applications

could play a major role in determining the competitiveness of low to moderate

renewable sources. An important aspect of the optimisation is to maximise the

turbine output power for a given resource. This requires careful attention to

the turbine design notably through numerical simulations. Challenges in the

numerical modelling of radial-inflow turbines using high-density working fluids

still need to be addressed in order to improve the turbine design and better

optimise ORCs. This paper presents preliminary 3D numerical simulations of a

high-density radial-inflow ORC turbine in sensible geothermal conditions. Fol-

lowing extensive investigation of the operating conditions and thermodynamic

cycle analysis, the refrigerant R143a is chosen as the high-density working fluid.

The 1D design of the candidate radial-inflow turbine is presented in details. Fur-

thermore, commercially-available software Ansys-CFX is used to perform pre-

liminary steady-state 3D CFD simulations of the candidate R143a radial-inflow

turbine for a number of operating conditions including off-design conditions.

The real-gas properties are obtained using the Peng-Robinson equations of state.

The thermodynamic ORC cycle is presented. The preliminary design created

using dedicated radial-inflow turbine software Concepts-Rital is discussed and

the 3D CFD results are presented and compared against the meanline analysis.

∗Corresponding authorEmail address: emilie.sauret@qut.edu.au (Emilie Sauret, Yuantong Gu)

Preprint submitted to Applied Energy August 24, 2014

Keywords: Radial-Inflow Turbine, Organic Rankine Cycles, Geothermal

Energy, CFD

Nomenclature

C0 [m.s−1] spouting velocity

cp0[kJ.kg−1.K] zero pressure ideal gas specific heat capacity

DRin[m] inlet rotor diameter

h [kJ.kg−1] enthalpy

M [-] absolute Mach number

Ns [-] stator number of blades

Nr [-] rotor number of blades

Pc [kPa] absolute pressure at the critical point

P [kPa] pressure

Qm [kg.s−1] mass flow rate

R [J.mol−1.K−1] ideal gas constant

Re [-] Reynolds number

Tc [K] absolute temperature at the critical point

T [K] temperature

U [m.s−1] blade velocity

y+w [-] non-dimensional grid spacing at the wall

W [kW] power

Wc [kW] cycle power

2

Greek Symbols

β [deg] blade angle

η [%] efficiency

ω [-] acentric factor

Ω [rpm] rotational speed

Π [-] pressure ratio

µ [kg.m−1.s−1] dynamic viscosity

Subscripts

in turbine inlet

is isentropic

out turbine outlet

ref reference

T Total

S Static

T − S Total-to-Static

T − T Total-to-Total

1. Introduction

Organic Rankine Cycles (ORCs) have been receiving lots of attention for

many years[1]. This interest is continuously growing [2] because of the suit-

ability of ORCs to low-to-mid-temperature renewable applications. Two key

parameters for the optimization of the ORC performance are the working fluid

and the turbine [3]. Extensive literature has been published on the working fluid

selection [4, 5, 6, 7, 8, 9] and separately on the preliminary one-dimensional tur-

bine design [10, 11, 12, 13, 14]. As underlined by Aungier [15], the preliminary

design of a radial-inflow turbine remains a crucial step in the design process.

Indeed, the objective of the design phases (preliminary and detailed) is to cre-

ate an aerodynamic design that will correctly and completely achieve a design

3

Table 1: Numerical simulations of turbines working with high-density fluids.

Author Working Fluid Simulation GeometryHoffren et al. [19] Toluene 2D Viscous radial turbine nozzleCinnella and Congedo [20] BZT Flows 2D Inviscid NACA Airfoil

transonic VKI LS-59linear turbine cascade

Boncinelli et al. [21] R134a 3D Viscous transonic centrifugalimpeller

Colonna et al. [22] Siloxane MDM 2D Inviscid radial turbine nozzleHarinck et al. [2] Toluene 2D Inviscid radial turbine nozzleHarinck et al. [23] Steam 2D Inviscid radial turbine nozzle

R245faToluene

Pini et al. [24] Siloxane MDM Pseudo-3D 3-stage and 6-stageThroughflow centrifugal turbine

Pasquale et al. [16] Not 2D Inviscid impulse radial turbineMentioned nozzle

Harinck et al. [18] Toluene 3D Viscous radial-inflow turbineWheeler and Ong [25] Pentane Quasi-3D radial turbine nozzle

R245fa ViscousKlonowicz et al. [26] R227ea 3D Viscous impulse turbineCho et al. [27] R245fa 3D Viscous impulse radial turbine

vaneless nozzle

that delivers the desired outputs. However, Pasquale et al. [16] also highlighted

the need to use more advanced 3D viscous computational simulations to further

investigate the proposed turbine configuration. Computational Fluid Dynam-

ics (CFD) will provide detailed flow analysis which is required to improve the

aerodynamic performance of the machine [17].

In order to achieve better cycle performances, the organic working fluid at the

inlet of the turbine is in thermodynamic conditions close or above to the critical

point. This prevents the numerical simulations to use ideal gas laws and so

require the developement of codes to calculate the thermodynamic properties of

the real gas. For accurate viscous simulations these codes need to be interfaced

with the CFD solvers [18].

Table 1 gives a summary of the published numerical simulations of expanders

working with organic fluids.

4

Since the end of the 90s, CFD solvers capable to handle real gas have been

developed [28, 20]. Then, numerical simulations on expanders and turbine noz-

zles using high-density fluids have been reported in the literature. In 2002,

Hoffren et al. [19] are amongst the first to perform two-dimensional numerical

simulations of a real gas flow in a supersonic turbine nozzle using toluene as

working fluid. Their extension of the Navier-Stokes solver for real gas appears

to be robust and accurate. Later, Colonna et al. [22] investigated the influence

of three different equations of state (EoS), a simple polytropic ideal gas law, the

Peng-Robinson-Stryjek-Vera cubic equation of state (EoS) and the state-of-the-

art Span-Wagner EoS, on the aerodynamic performance of a two-dimensional

ORC stator blade working with the siloxane MDM. The two more sophisticated

models (Peng-Robinson-Stryjek-Vera and Span-Wagner) were shown to produce

similar results. Pini et al. [24] recently performed a 2D throughflow simulation

on their proposed centrifugal turbine for an ORC application working with the

siloxane MDM. Finally, to the best of the authors’ knowledge, Boncinelli et

al. [21] in 2004 were the first to successfully perform three-dimensional viscous

simulations of a transonic centrifugal impeller working with R134a using a fit-

ting model from gas databases to get the thermodynamic properties of the fluid.

Recently, pseudo-3D CFD simulations using equations of state for the real dense

gas are produced by Wheeler and Ong [25] on a turbine vanes. Finally, Harinck

et al. [18] were the first to perform three-dimensional viscous CFD simulations

of a complete radial turbine including the stator, the rotor and the diffuser using

toluene as the working fluid.

Pasquale et al. [16] looked at the blade optimisation of expanders using real

gas. However, only two-dimensional nozzle simulations have been considered.

Recent studies [26, 27] have looked at the whole process including thermo-

dynamic cycle, 1D design and numerical simulations of the expander. Klonow-

icz [26] designed and numerically investigated a R227ea impulse turbine at par-

tial admission. However, they only simulate one stator and rotor pitch and so

weren’t able to take numerically into account the partial adnmission. Both cases

with and without clearances were run. Cho et al. [27] also numerically studied

5

a partial admission radial impulse turbine with a vaneless nozzle. The ouput

power in this case is 30kW and the working fluid R245fa.

It is important to note that so far, very few studies including the whole

process for sensible geothermal conditions have been presented in the literature.

Furthermore, 3D viscous simulations on complete turbines working with organic

fluids are at a pioneering stage and all the published numerical studies on ex-

panders coupled with ORCs consider small-scale power cycles with output power

less than 50kW. Most of them are impulse turbines with partial admission.

This paper will fill that gap by providing a complete ORC analysis including

the thermodynamic cycle, the preliminary meanline design and finally pioneer-

ing three-dimensional CFD simulations of a 400kW-R143a radial-inflow turbine

in sensible geothermal brine and cycle conditions. Off-design conditions are also

presented and discussed.

2. Organic Rankine Cycle

A thermodynamic model for a binary power cycle was developed using En-

gineering Equation Solver (EES, F-Chart Software) based on the basic compo-

nents of an ORC system: evaporator, turbine, generator, condenser and fluid

pump (Figure 1). Simulations for realistic condensing ORC cycles for geother-

mal applications were carried out and analysed leading to the promising cycle for

a brine temperature of 423K at a sensible flow rate of 15kg.s−1 as encountered

during the testings at the Habanero pilot plant in Australia. The condensing

temperature was set at 313K (10K above ambient, assumed to be 303K) while

the turbine inlet temperature is fixed to 413K through a temperature difference

of 10K with the brine inlet temperature at 423K. In this analysis, turbine and

pump efficiencies were assumed to be 85% and 65%, respectively. Promising

cycles were identified by varying the evaporator pressure in order to optimise

the net cycle power. As outlined by Sauret and Rowlands [29] and recently by

Toffolo et al. [9] a coupled optimisation between the thermodynamic cycle and

the cycle components such as the 1D turbine design is required to maximize

6

Figure 1: Basic Organic Rankine Cycle (ORC) System.

the net cycle power. Such a coupled Multi-Objective optimisation has been

developed by Persky et al. [30] and applied to the present study. Therefore,

the optimum turbine inlet total pressure has been optimised to 5MPa which

lead to a performance factor (PF = Wc/D2Rin

) of 20.6 compared to PF=17.2

previoulsy obtained by Sauret and Rowlands [29].

In this study, the refrigerant R143a is chosen as the working fluid for the

turbine based on the study of Sauret and Rowlands [29]. This study evalu-

ated 5 different fluids based on the QGECE (Queensland Geothermal Energy

Center of Excellence) thermodynamic modelling and published results from oth-

ers [4, 7, 31, 32, 33]. It showed that R143a at 413K gave the best performance

factor based on the cycle net power and the size of the turbine. Thus, R134a

is a suitable fluid at 413K capable to generate a relatively large turbine power

compared to the compactness of the turbine. The choice of R143a is based not

only on its theoretical thermodynamic performance, but a balance of several

different desired criteria such as relatively low critical pressure and temperature

and low flammability, low toxicity and relative inert behaviour. Striking an ac-

7

ceptable balance between the aforementioned criteria, environmental concerns,

cost and thermodynamic performance is not an insignificant challenge for ORC

system designers [4].

The selection of the radial-inflow turbine is based on numerous published

studies highlighting the suitability of such turbines for ORC cycles [34, 29, 35,

24]. As mentioned by Clemente et al. [36], the selection of the best suited

machine requires a detailed analysis of the given case.

The thermodynamic data required for the turbine design are summarized

table 2.

Table 2: Thermodynamic data required for the turbine design obtained from

the R143a ORC cycle analysis.

Variables PTin [kPa] PSout [kPa] TTin [K] W [kW]

5000 1835 413 400

3. One-dimensional design of the radial-inflow turbine

In this study, the preliminary design of the turbines was performed using

the commercial software RITAL following the design procedure established by

Moustapha et al. [37]. The general procedure implemented was to determine

overall dimensions of the machine for the stator, rotor and diffuser along with

blade and flow angles and isentropic efficiency by fixing the mass flow rate, inlet

and outlet pressures as calculated by the cycle analysis (Table 2) and by assum-

ing flow and loading coefficients [37]. From the basic design specifications and

the performance analysis, a refinement of the results was performed in order to

optimise the geometry. The refinements included: rotational speed adjustments

in order to get closer to the optimum 0.7 value of the U/C0 ratio [37]; tip outlet

rotor radius modification in order to reduce the swirl as much as possible at the

outlet of the rotor; and, blade span (i.e. changes to throat area).

8

Table 3: Preliminary R143a Turbine Design Data.

Global Variables Geometric Parameters

ΠT−S [-] 2.72 Ns [-] 19

ΠT−T [-] 2.62 Nr [-] 16

Qm [kg/s] 17.24 DRin[mm] 127.17

ηT−S [%] 76.8

ηT−T [%] 79.8

Ω [rpm] 24250

W [kW] 393.6

U/C0 [-] 0.675

Assumptions and constraints. In the Moustapha et al. procedure [37], the flow

and loading coefficients are assumed in order to give the best efficiency. In this

current study, these coefficients were set respectively to 0.215 and 0.918 giving

around 90% efficiency according to the Flow and Loading Coefficient Correlation

Chart proposed by Baines [13]. The rotor clearances were set using an assumed

value of 0.3 mm. The wish to use widely available inverter technology in order

to simplify future experimental rigs led to the shaft speed being limited to a

maximum of 25,000 rpm. The default RITAL recommended loss models were

used: the Rodgers stator loss model [38] and the CETI rotor passage loss model

from Concepts [39]. These models were found to provide a reasonable first

approximation.

1D design results. The results of the preliminary design are presented Table 3.

The power is slightly lower than the expected value from the thermodynamic

cycle. However, the total-to-total efficiency almost reached 80% while the total-

to-static efficiency is close to 77%. This is in close agreement with the previous

study proposed by Sauret and Rowlands [29]. The size of the turbine is also in

close agreement with the previous study [29]. One can also note that because of

a relatively high power compared to a relatively small size, this turbine achieves

a high performance factor (PF = Wc/D2Rin

) of 20.6 compared to PF=17.2

9

obtained by Sauret and Rowlands [29].

4. 3D Geometry

The commercial software Axcent were used to define the 3D geometry of the

turbine presented in Figure 2.

Figure 2: 3D view of the full 3D turbine geometry (stator and rotor blades).

The data from the preliminary 1D design were transferred into Axcent, in-

cluding amongst the most important parameters, radii and blade heights. An

annular diffuser at the exit of the rotor is built from the 1D radii values. No

optimisation was carried out on the 3D diffuser geometry. The nozzle hub and

shroud contours are defined as straight lines. The nozzle hub and shroud thick-

nesses, the rotor blade angle and rotor thickness distributions were iteratively

adjusted using Bezier curves in order to achieve reasonnable preliminary results

with blade-to-blade and throughflow solvers notably in terms of loading and

Mach numbers which were maintained in the transonic region. The same thick-

ness distribution was used for the stator hub and shroud while two separate

distributions were set for the rotor hub and shroud.

10

The main 3D geometric parameters are presented in Table 4.

Table 4: Turbine 3D Geometric Dimensions

Nozzle Rotor Diffuser

Rin [mm] 83.45 Rin [mm] 63.6 Rtipin [mm] 44

Blade Height [mm] 8 Inlet Blade Height [mm] 8 Rhubin [mm] 18.5

TE Thickness [mm] 1 TE Thickness [mm] 1.27

Rout [mm] 66.76 Axial Length [mm] 38.15 Axial Length [mm] 45

βtipin [deg] 0

βhubin [deg] 0

βtipout [deg] 60.4

βtipin [deg] 38.9

Clearances [mm]:

Axial 0.3

Radial 0.3

5. Three-Dimensional Numerical Simulations

Viscous turbulent CFD simulation was carried out on the 3D turbine geom-

etry at the nominal condition using commercially-available software ANSYS-

CFX.

Mesh. The O-H three-dimensional computational mesh for the simulation of

one blade passage is shown in Figure 3 for the nozzle and in Figures 4 and 5 for

the rotor shroud and hub respectively.

An initial grid was created with a total grid number is 1359907 nodes, 678,389

for the stator, 444,341 for the rotor and 237,177 for the diffuser. The average

non-dimensional grid spacing at the wall y+w = 703 is slightly above the recom-

mended range of up to 500. With the scalable wall function used in the simu-

lations, recommended y+w values lay between 15 and 100 for machine Reynolds

number of 105 where the transition is likely to affect significantly the boundary

layer formation and skin friction and up to 500 for Reynolds number of 2× 106

when the boundary layer is mainly turbulent throughout [40]. However, based

11

Figure 3: 3D view of the O-H grid around the stator blades.

on Remachine = (ρrefUtipRin

RRin)/µref where µref = 1.9 × 105[kg/(m.s)] is the

reference dynamic viscosity of R143a and ρref = 163.2[kg/m3] the reference

density at the inlet of the domain, the present machine Reynolds number at the

inlet is 8.8× 107 well above 2× 106 . This initial mesh was found good enough

to run pioneering simulations of the 3D candidate R143a radial-inflow turbine

in sensible geothermal brine and cycle conditions and establish the complete

methodology (thermodynamic cycle, meanline analysis and preliminary viscous

CFD simulations).

Numerical Model. The CFD solver ANSYS-CFX has been used to perform

steady-state 3D viscous flow simulations. For robustness consideration, the

discretization of the Navier-Stokes equations is realized using a first order Up-

wind advection scheme. For these preliminary 3D simulations, the standard k-ε

turbulence model with scalable wall function was chosen.

12

Figure 4: 3D view of the O-H grid around the rotor blade at the shroud.

Figure 5: 3D view of the O-H grid around the rotor blade at the hub.

In order to describe the properties of the refrigerant R143a in the radial-inflow

turbine, the cubic equation of state (EoS) of Peng-Robinson [41] (Equation 1)

is chosen. This choice was motivated by the good balance between simplicity

and accuracy of this model to perform preliminary real gas computational sim-

ulations. This EoS is also considered as having reasonable accuracy especially

near the critical point [42].

13

p =RT

Vm − b− aα

V 2m + 2bVm − b2

a =0.457235R2T 2

c

pc

b =0.077796RTc

pc

α =(1 + κ(1− T 0.5r ))2

κ =0.37464 + 1.54226ω − 0.26992ω2

Tr =T

Tc

(1)

where ω = 0.259 is the acentric factor of refrigerant R143a, Tc = 346.3 K and

Pc = 3492 kPa are respectively the critical temperature and pressure of R143a,

Vm, the molar volume and R is the universal gas constant.

To complete the description of the real gas properties, the CFD solver calcu-

lates the enthalpy and the entropy using relationships which are detailed in

[43]. These relationships depend on the zero pressure ideal gas specific heat

capacity cp0and the derivatives of the Peng-Robinson EoS. cp0

is obtained by

a fourth-order polynomial whose coefficients are defined by Poling et al. [44].

Boundary Conditions. An extension of the domain was placed in front of the

stator blades at a distance equivalent to approximately 25% of the axial stator

chord. The nominal rotational speed for this case is defined from the preliminary

design and set to 24,250 rpm (Table 3). The mass flow rate, total temperature

and inlet velocity obtained from the meanline design analysis were set at the

inlet of the domain (Table 2). The static pressure was specified at the exit of

the diffuser (Table 2).

A mixing plane condition was set at the interface between the stationary (stator)

and rotational (rotor) frames and a frozen rotor interface was set between the

rotor and the diffuser. Periodic boundary conditions are applied as only one

blade passage for both the stator and the rotor is modelled.

14

Table 5: Comparison of the meanline analysis and the present 3D CFD simula-

tions.

Variables Preliminary Design Present CFD

Global Variables

ΠT−S [-] 2.72 2.64

ΠT−T [-] 2.62 2.54

ηT−S [%] 76.8 83.5

ηT−T [%] 79.8 87.6

W [kW] 393.6 421.5

Stator Inlet

hT [kJ.kg−1] 499.2 501

PT [kPa] 4989.6 4849.85

Rotor Inlet

M [-] 0.88 0.94

PT [kPa] 4886.5 4717

TT [K] 412.2 411.9

Rotor Outet

M [-] 0.363 0.364

PT [kPa] 1961.4 1933.1

TT [K] 370.2 367.1

hT [kJ.kg−1] 476.4 476.3

6. Results and Discussion

Comparisons of the 3D CFD results against the meanline analysis at the nominal

condition. Due to the lack of experimental data in radial-inflow turbines work-

ing with high-density fluids, the validation of the present 3D viscous simulation

was firstly made against the meanline analysis results.

Table 5 compares the results from both the preliminary 1D design and the

numerical 3D turbulent simulations. It is observed that the CFD results are

in relatively good agreement with the 1D analysis, especially in terms of tem-

peratures, pressures and enthalpy. However, the slight difference (less than

0.5%) on the total enthalpy at the inlet of the nozzle and outlet of the rotor

15

between the viscous simulation and the 1D analysis leads to approximately 8%

difference in terms of ∆hT = hTin− hTout

explaining the difference of power as

W = Qm(hTin− hTout

).

The CFD results are in fairly good agreements with the meanline analysis except

for efficiencies for which a maximum difference of nearly 9.8% is obtained. The

overestimation of the efficiency has already been highlighted in a less extent by

Sauret [45] on the simulations of an ideal gas high pressure ratio radial-inflow

turbine. The present result can be attributed to the use of real gases which

are extremely sensitive to small temperature variations. As shown in Table 5,

a slight variation of the total temperatures and enthalpies can lead to large

variation of the enthalpy and temperature drops in the turbine. The isentropic

efficiencies defined below in Equation 2 are function of the enthalpy drop:

ηT−T =hTin − hTout

hTin− hTisout

ηT−S =hTin − hTout

hTin− hSisout

(2)

Thus, any small variations of temperatures can lead to big discrepancy in

efficencies. Several studies [46, 42] have shown that the Peng-Robinson EoS was

comparing fairly well against the high-accuracy models implemented in REF-

PROP [47] and especially near the critical point. Thus, it is not assured that

a more accurate EoS will dramatically improve the CFD efficiencies. Instead,

it is highly recommended here to investigate the sensitivity of the efficiency to

the EoS using non-statistical state-of-the-art uncertainty quantification methods

such as Stochatic Collocation using Adaptive Sparse Grid.

Furthermore, compared to the meanline design, the Mach number at the

inlet of the rotor (Table 5) is also higher. The Mach number at mid-span is

presented Figure 6.

One can note that the exit of the stator is slightly chocked with a maximum

Mach number of approximately 1.05. This, in return, leads to the higher Mach

number at the inlet of the rotor. As a consequence, further 3D shape optimisa-

tion is required in order to improve the blade loading profile shown in Figure 7

and thus reduce the Mach number in order to remain in the transonic region.

16

Figure 6: Mach number at mid-span for the nominal condition.

Also the relatively high loading at the leading edge of the rotor presented Fig-

ure 7 associated with relatively low loading from the mid-blade to the trailing

edge may cause structural issues for the blade integrity. This will require in

the future to also run a structural blade analysis which would need to be linked

to the blade optimisation. Thus, it is very likely that the CFD simulations

on a fully-optimized 3D geometry with a more accurate modelling of the real

gas properties will provide much better agreement with the meanline design in

terms of efficiencies.

Results at off-design conditions. The 3D numerical simulations have also been

performed at off-design conditions for different rotational speeds, inlet total

temperatures and pressure ratios. The evolution of the total-to-total efficiency

with the total inlet temperature and rotational speed are presented in Figure 8

and Figure 9 respectively.

As one can note in Figure 8, at the nominal rotational speed, the turbine

can effectively handle temperature variations with a maximum efficiency differ-

ence of 3.3% over the range of tested temperatures. The maximum efficiency

is reached at TTin=405K, slightly lower than the nominal temperature of 413K.

17

Figure 7: Blade loading at three different rotor blade heights for the nominal

condition.

0.6

0.65

0.7

0.75

0.8

0.85

0.9

380 400 420 440 460

ηT

-T [-]

TTin [K]

80% RPMNominal RPM

120% RPM

Figure 8: Evolution of the total-to-total efficiency with the total inlet tempera-

ture at three different rotational speeds.

At 80% rpm, the total-to-total efficiency continually decreases with the inlet

temperature while at the highest rotational speed, the efficiency increases with

the temperature and reaches a plateau from TTin=430K. The total-to-total ef-

ficiency dramatically drops at high rotational speed with low temperatures.

Figure 9 shows that at low temperature, the maximum efficiency is reached

18

0.6

0.65

0.7

0.75

0.8

0.85

0.9

70 80 90 100 110 120 130

ηT

-T [-]

Ω/ΩNominal [%]

TTin=380K

TTin=413K

TTin=450K

Figure 9: Evolution of the total-to-total efficiency with the rotational speed at

three different inlet temperatures.

at a low rotational speed. After that point, the total-to-total efficiency rapidly

drops with an increase of the rotational speed. At the nominal temperature

(413K), the turbine can better handle the variation of rotational speed. In

that case, the maximum efficiency is reached at 105% of the nominal rotational

speed (25,463.5 rpm), slightly above the nominal rotational speed of 24,250 rpm.

At higher temperature, the efficiency keeps increasing with the rotational speed.

Figure 10 and Figure 11 presents the variation of power with the inlet tem-

perature and the rotational speed respectively.

For all rotational speeds, the power increases with the inlet turbine temperature

as shown in Figure 10. However, this increase is more pronounced at the highest

rotational speed. It is also observed in Figure 11 that, as the temperature

increases, the maximum power tends to move towards higher rotational speeds.

Furthermore, the power increases with the temperature for a constant rotational

speed.

A variation of the pressure ratio was also performed and the results at three

different inlet temperatures for the nominal rotational speed are presented in

Figure 12. At the lowest temperature, the maximum efficiency is reached at

ΠT−S=2.3. The efficiency rapidly drops on each side of the best efficiency point

19

Figure 10: Variation of the turbine power with the total inlet temperature at

three different rotational speeds.

Figure 11: Variation of the turbine power with the rotational speed at three

different inlet temperatures.

such that no numerical results were obtained for pressure ratios above 2.9 at 80%

of the nominal rotational speed. For the nominal temperature (TTin=413K), the

maximum efficiency is reached at ΠT−S=2.1 well below the nominal pressure

ratio of 2.65. After this point, the efficiency gradually decreases with an increase

of the pressure ratio. Similar trends are observed at the highest temperature

with a faster drop compared to the nominal temperature.

The evolution of the efficiency with the pressure ratio was also plotted at

20

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

ηT

-T [-]

ΠT-S [-]

TTin=380K

TTon=413K

TTin=450K

Figure 12: Variation of the total-to-total efficiency with the total-to-static pres-

sure ratio at three different inlet temperatures for the nominal rotational speed.

three different rotational speeds as shown in Figure 13 at the nominal inlet

temperature.

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

ηT

-T [-]

ΠT-S [-]

80% RPMNominal RPM

120% RPM

Figure 13: Variation of the total-to-total efficiency with the total-to-static pres-

sure ratio at three different rotational speeds for the nominal inlet temperature.

At low rotational speed, the efficiency continually decreases with the total-to-

static pressure ratio while at the nominal rotational speed, the maximum effi-

ciency is reached at ΠT−S=2.1 and then decreases. At the highest rotational

speed, the maximum efficiency is shifted towards higher total-to-static pressure

ratio (2.9). At low total-to-static pressure ratio, the highest rotational speed

21

gives the lowest efficiency while at higher pressure ratios, it is the lowest rota-

tional speed that provides with the lowest efficiency. The highest efficiencies are

obtained over the range of pressure ratios at the nominal rotational speed.

Discussion. We can note that the effects of rotational speed, inlet temperature

and pressure ratio are less pronounced at the nominal conditions. Therefore, the

nominal point is the most capable of handling a variation of rotational speed,

temperature or pressure ratio while maintaining a relatively high efficiency over

the range of variation. The effect of the rotational speed is also more impor-

tant at TTin=380K than at TTin

=450K. This is explained by the variation of

enthalpy drop through the turbine as shown in Figures 14 and 15. We can note

that at TTin=380K, the enthalpy drop dhT at 120% rpm is 32.1% smaller than

at 80% rpm. Thus the efficiency at TTin=380K is largely higher for the lowest

rotational speed. On the contrary, at TTin=450K, dhT at 120% rpm is 17.1%

larger than dhT at 80% rpm explaining the lower efficiency at TTin=450K for

the lowest rotational speed. In addition, as shown in Figure 16, increasing the

rotational speed at the lowest temperature dramatically damage the flow with

large separation areas in the rotor associated with large density values at the

exit of the stator. The effect of the rotational speed at the highest temperature

is different as the flow pattern improves with the increase of rotational speed. At

TTin=450K and 120% rpm no separation is visible at mid-span. Furthermore,

the flow at these conditions is really similar to the flow at TTin=380K, 80%

rpm explaining the similar efficiencies obtained for these 2 conditions (Figures 8

and 9). All these effects are related to the high-density fluid properties asso-

ciated with opposite flow behaviours when increasing the temperature at 80%

rpm and 120% rpm or when increasing the rotational speed at TTin=380K and

TTin=450K. Thus accurate advanced uncertainty quantification would benefit

the understanding of the sensitivity of the numerical simulations to the fluid

properties.

The variation of the power presented in Figures 10 and 11 is also linked to the

fluid properties, enthalpy drop as seen in Figure 14 and 15 and flow modifications

22

1.0e4

1.5e4

2.0e4

2.5e4

3.0e4

3.5e4

70 80 90 100 110 120 130

dh

T [

kJ.k

g-1

]

Ω/ΩNominal [%]

TTin=380K

TTin=413K

TTin=450K

Figure 14: Evolution of the enthalpy drop with the rotational speed at 3 different

inlet temperatures.

1.0e4

1.5e4

2.0e4

2.5e4

3.0e4

3.5e4

380 400 420 440 460

dh

T [

kJ.k

g-1

]

TTin [K]

80% RPMNominal RPM

120% RPM

Figure 15: Evolution of the enthalpy drop with the inlet temperature at 3

rotational speeds.

in the turbine (Figure 16). The variation of enthalpy drop through the turbine

is different according to the temperature. While dhT keeps increasing with

the rotational speed at 450K, it starts decreasing for 380K after 95% rpm due

to the increased separation regions in the rotor with the rotational speed at

the lowest temperature. This low temperature high rotational speed case also

23

corresponds to a very high density bringing back the question of the sensitivity

of the Peng-Robinson EoS. Thus further detailed uncertainty quantification is

required to accurately quantify this sensitivity.

(a) TTin=380K, 80% nominal rpm (b) TTin=450K, 80% nominal rpm

(c) TTin=380K, 120% nominal rpm (d) TTin=450K, 120% nominal rpm

Figure 16: Mach number at mid-span for TTin=380K and TTin

=450K at 80%

and 120% nominal rpm.

The nominal point and all the off-design conditions numerically simulated

are gathered in Figure 17 which shows the variation of efficiency with the blade-

to-jet speed ratio U/C0.

One can note that the maximum efficiency of 88.8% is reached at a blade-to-

jet speed ratio of 0.748 well above the optimum value of 0.7 [11, 13]. However,

this point corresponds to a turbine output power of only 257kW well below

the nominal power of around 400kW. For that reason, the efficiency versus the

blade-to-jet speed ratio was plotted only considereing the points with an output

power of approximately 400kW. The resulting graph is shown in Figure 18.

While only considering the conditions leading to the nominal turbine power of

24

0.4

0.5

0.6

0.7

0.8

0.9

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

ηT

-T [-]

U/C0 [-]

Figure 17: Variation of the total-to-total efficiency with the blade-to-jet speed

ratio.

400kW, one can note that the maximum efficiency is now 88.45% and is reached

at U/C0=0.711, slightly above the optimum value of 0.7.

The nominal point is then compared to the best efficiency points in Table 6.

Table 6: Comparison of the nominal best efficiency points.

Variables Nominal Point Best Efficiency Point Best Efficiency Point

at W=400kW

ηT−T [%] 87.6 88.8 88.45

ΠT−S [-] 2.64 2.1 2.46

TTin[K] 413 413 413

Ω [rpm] 24250 24250 24250

Qm [kg.s−1] 17.24 12.56 17.24

W [kW] 421.5 257.2 390.8

U/C0 [-] 0.681 0.748 0.711

The best efficiency point is almost the efficiency obtained at a fixed turbine

power of approximately 400kW. Furthermore, the nominal efficiency is ex-

tremely close to the best efficiency with a 1.4% difference. We can also note that

the best efficiencies are obtained at the nominal inlet temperature and nominal

rotational speed. However, the best efficiency is obtained at a lower pressure

ratio due to a lower mass flow rate, In return, the power is lower. At the nominal

25

0.8

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

0.89

0.9

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

ηT

-T [-]

U/C0 [-]

Figure 18: Variation of the total-to-total efficiency with the blade-to-jet speed

ratio at nominal turbine power W = 400kW .

turbine power, the best efficiency is obtained at the nominal flow rate but also

with a reduced pressure ratio, achieved by varying the outlet static pressure. A

coupled optimisation of the thermodynamic cycle and 1D turbine design would

be then beneficial. It would provide the best outlet pressure for the turbine in

order to achieve the the best efficiency at the required turbine power output.

7. Conclusion

This paper is the first to present the full design process of a radial-inflow

turbine working with R143a in sensible geothermal brine and cycle conditions.

The thermodynamic cycle, the preliminary 1D design, the meanline analysis of

the turbine and eventually the steady-state 3D viscous simulations of the tur-

bine both at nominal and off-design conditions are produced.

The CFD simulations provided satisfactory results compared to the meanline

analysis except for the efficiencies due to the sensitivity of real gas properties

to the temperature variations. Parametric studies were carried out highlighting

the relative robustness of the design point over the variations of temperature,

rotational speed and pressure ratio. The CFD results also confirmed the need

for a coupled optimisation of the thermodynamic cycle and 1D turbine design.

Optimising the system parameters will positively impact the turbine and thus

26

cycle efficiencies.

These pioneering 3D CFD simulations confirmed previously-published studies

for the need of more detailed optimization in order to improve the blade load-

ing profiles of the turbine and lower the maximum Mach number at the exit

of the nozzle. As a perspective to the presented work, the direct use of the

RefProp database for the real gas properties in the CFD solver instead of the

Peng-Robinson EoS will be performed in order to improve the prediction of the

thermodynamic properties and so the power and turbine efficiency. In addition,

it is recommended that the sensitivity of the efficiency to the EoS will be studied

using non-statistical state-of-the-art uncertainty quantification methods such as

Stochatic Collocation using Adaptive Sparse Grid.Finally, a 3D shape optimiza-

tion tool coupled with a structural analysis will be developed and applied to this

candidate R143a radial-inflow turbine in sensible geothermal conditions.

Acknowledgements

Dr. E. Sauret would like to thank the Australian Research Council for its

financial support

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