# Unidimensional and 3D Analyses of a Radial Inflow Turbine for an Organic Rankine Cycle under Design and Off-Design Conditions

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^{2}

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## Abstract

**:**

## 1. Introduction

_{2}H

_{5}-OH). The parameters of the preliminary turbine design were analyzed and calculated in MATLAB, while the 3D analysis of the turbine rotor and nozzle passage was performed in MATLAB ANSYS-CFX 14.5. The K-ω and k-ϵ shear stress transport (SST) turbulence models were implemented in this study. Miranda [31] presented a preliminary design of a subsonic radial turbine that works with organic fluids, such as R-123, R-134a, R-141b, and R-152a, that considered the real properties of these fluids. For these tests, the author used Fortran to develop the parameter calculation algorithm and CFD to perform the 3D analysis of the turbine. The program developed in Fortran for the one-dimensional analysis of the turbine determined the characteristics of the flow along the mean streamline for the point of operation under steady-state conditions. The parameters calculated by the one-dimensional analysis were used as inputs for the 3D design of the turbine in ANSYS. Under such conditions, the author ultimately found that R-141b enabled the largest turbine, with a rotor of 114.144 mm in radius and with a total efficiency of 70.504%. The analysis of turbine efficiency as a function of blade tip clearance and rotor backplate showed that the latter had a stronger effect on turbine operation and power output.

## 2. One-Dimensional Model of the Radial Turbine

## 3. Rotor Model

_{b4}), outlet blade tip thickness (t

_{6t}), outlet blade root thickness (t

_{6h}), outlet blade root radius (r

_{h6}), the axial length of the rotor (L

_{R}), and the number of blades (${\mathrm{Z}}_{\mathrm{R}}$) are determined using the following equations [42,47]:

_{6}) is calculated using Equation (30).

## 4. Nozzle Model

_{s}is a nozzle design parameter [1.0–2.8] [50] and ${C}_{s}$ is the nozzle blade chord. The nozzle blade chord (${C}_{s}$) is calculated using Equation (43) [50].

## 5. Volute Model

## 6. Radial Turbine Losses

#### 6.1. Rotor Losses

#### 6.1.1. Friction Losses

#### 6.1.2. Outlet Losses

#### 6.1.3. Aerodynamic Load Losses

#### 6.1.4. Trailing Edge Loss

#### 6.2. Nozzle Losses

#### 6.3. Volute Losses

## 7. 3D Radial Turbine Design

^{®}, the meshes were generated and configured in ANSYS Turbogrid

^{®}, and the boundary conditions of each case analyzed and simulated in this study were defined using the tool CFX-Pre

^{®}. In addition, the simulation control conditions were set in CFX-Pre

^{®}for each case developed in this study. After all the steps shown in Figure 7, the 3D geometries of the ORC turbine nozzle and radial rotor were obtained based on the working fluid and operating conditions defined for each project. The simulations performed in the module CFX-Pre

^{®}yielded pressure, velocity, and Mach number profiles across turbine components.

## 8. Validation of One-Dimensional Mathematical Models of the Radial Turbine

## 9. Radial Turbine Design Results

## 10. Efficiency Characteristics of the Radial Turbine Nozzle-Rotor Set

## 11. Characteristic Curves of Radial Turbines

## 12. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | Area (m^{2}) |

b | Normal, interaction coefficient, base, nozzle blade |

C | Specific heat (kJ/kg·K), Absolute velocity |

${C}_{s}$ | Nozzle Chord |

d | Diffuse |

$D$ | Diameter (m) |

eva | Evaporator |

$f$ | Friction factor, fluid |

F_{t} | Correction factor |

$h$ | hub |

${k}_{volute}$ | Pressure loss coefficient |

L | Length (m) |

m | Meridional plane |

${\dot{m}}_{}$ | Mass flow (kg/s) |

p | Constant pressure, loss, projection |

R | Rotor blade |

r | Radius (m) |

rms | Root mean square radius |

$T$ | Temperature (K) |

s | shroud, storage, dry section |

${S}_{s}$ | Nozzle design parameter |

U | Circumferential rotation speed (m/s) |

v | Volute, Volume (m^{3}) |

W | Relative velocity (m/s) |

$\mathrm{Z}$ | Nozzle blades number |

$\omega $ | Rotation speed (rad/s) |

Greek symbols | |

$\alpha $ | Absolute flow angle (°) |

β | Relative flow angle (°) |

$\epsilon $ | Rotor radius ratio |

$\theta $ | Tangential plane |

γ_{fi} | Interception factor |

ξ | Meridional speed ratio |

$\psi $ | Loading coefficient |

Φ | Flow coefficient |

Subscribed | |

0 | Total, Volute input |

1 | Receiver side one, Nozzle inlet |

2 | Receiver side two |

3 | Nozzle outlet |

4 | Rotor inlet |

6 | Rotor outlet |

in | Inlet |

$ise$ | Isentropic |

$out$ | Outlet |

orc | Organic Rankine Cycle |

PTC | Parabolic Trough Collectors |

wf | Working fluid |

$sur$ | Surroundings |

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**Figure 2.**Meridional view of the radial inflow turbine, indicating different components and main dimensions, adapted from [41].

**Figure 3.**Velocity triangle at the rotor inlet of a radial turbine, adapted from [45].

**Figure 4.**Velocity triangle at the rotor outlet of a radial turbine, adapted from [44].

**Figure 5.**Simplified diagram of the nozzle adapted from [29].

**Figure 13.**Variation in the total isentropic efficiency and power of the turbine operating with R-245fa, as a function of pressure ratio.

**Figure 14.**Variation in the total isentropic efficiency and power of the turbine operating with R-245fa, as a function of rotation speed.

**Figure 15.**Variation in the total isentropic efficiency and power of the turbine operating with R-141b, as a function of pressure ratio.

**Figure 16.**Variation in the total isentropic efficiency and power of the turbine operating with R-141b, as a function of rotation speed.

**Figure 17.**Variation in the total isentropic efficiency and power of the turbine operating with R-123, as a function of pressure ratio.

**Figure 18.**Variation in the total isentropic efficiency and power of the turbine operating with R-123, as a function of rotation speed.

Radial Turbine Parameters | |
---|---|

Fluid | R-245fa |

Flow coefficient (ϕ) | 0.215 |

Pressure coefficient (ψ) | 0.918 |

Inlet total pressure (kPa) | 1000 |

Condenser working temperature (K) | 303.15 |

Rotation (rpm) | 9000 |

Mass flow rate (kg/s) | 10.92 |

Parameter | Paltrinieri [52] | NEST Model | Difference (%) |
---|---|---|---|

Rotor | |||

Inlet radius, r4 (m) | 0.1796 | 0.1709 | 4.8 |

Inlet blade height, b4 (m) | 0.0114 | 0.0106 | 7.0 |

Inlet absolute flow angle, α4 (°) | 76.8186 | 76.8186 | 0 |

Inlet relative flow angle, β4 (°) | −43.4682 | −42.5484 | 2.1 |

Inlet static temperature, T4 (K) | 354.62 | 353.7832 | 0.24 |

Inlet static pressure, p4 (kPa) | 585.37 | 590.7688 | 0.92 |

Inlet absolute Mach number, Ma4 | 1.0369 | 1.0479 | 1.06 |

Inlet relative Mach number, Ma4′ | 0.3258 | 0.2788 | 14.4 |

Outlet hub radius, r6h (m) | 0.0539 | 0.053 | 1.7 |

Outlet shroud radius, r6s (m) | 0.1169 | 0.1182 | 1.1 |

Outlet blade height, b6 (m) | 0.063 | 0.0653 | 3.7 |

Outlet absolute flow angle, α6 (°) | −0.4018 | −0.4018 | 0 |

Outlet relative flow angle, β6_rms | −64.0532 | −65.0962 | 1.6 |

Outlet static temperature, T6 (K) | 325.74 | 323.8017 | 0.59 |

Outlet static pressure, p6 (kPa) | 171.81 | 175.12 | 1.92 |

Outlet absolute Mach number, Ma6 | 0.2582 | 0.2434 | 5.7 |

Outlet relative Mach number, Ma6´ | 0.59 | 0.6277 | 4.7 |

Number of blades, Z_{R} (−) | * | 18 | |

Nozzle | |||

Inlet radius, r1 (m) | 0.2335 | 0.2222 | 4.8 |

Inlet blade height, b1 (m) | 0.0114 | 0.0106 | 7.0 |

Inlet static temperature, T1 (K) | 369.75 | 369.89 | 0.04 |

Inlet static pressure, p1 (kPa) | 995.88 | 999.9 | 0.4 |

Inlet absolute Mach number, Ma1 | 0.095 | 0.1100 | 15.7 |

Outlet radius, r2 (m) | 0.1811 | 0.1734 | 4.3 |

Outlet blade height, b2 (m) | 0.0114 | 0.0106 | 4.38 |

Outlet absolute flow angle, α2 (°) | 76.8186 | 76.8186 | 4.14 |

Inlet static temperature, T2 (K) | 354.62 | 353.7832 | 0.174 |

Outlet static pressure, p2 (kPa) | 585.37 | 590.7688 | 0.107 |

Outlet absolute Mach number, Ma2 | 1.0369 | 1.0479 | 1.06 |

Number of blades, Z_{B} (−) | 25 | 25 | 0 |

Total to static efficiency, η_{ts} (%) | 74.1 | 71.85 | 3.0 |

Net Power, P (kW) | 267 | 260.5 | 2.4 |

Parameter/Fluid | R-245fa | R-141b | R-123b |
---|---|---|---|

Turbine nozzle inlet total temperature (°C) | 79.9 | 94.3 | 96.6 |

Mass flow rate of the working fluid (kg/s) | 0.61 | 0.45 | 0.62 |

Flow coefficient (ϕ) | 0.215 | 0.215 | 0.215 |

Load coefficient (ψ) | 0.918 | 0.918 | 0.918 |

Pressure ratio | 5.71 | 4.71 | 4.71 |

Rotation speed, ω (rpm) | 25,800 | 22,000 | 24,000 |

Parameter | R-245fa | R-141b | R-123 |
---|---|---|---|

Rotor | |||

Inlet radius, r_{4} (m) | 0.0568 | 0.0592 | 0.0531 |

Inlet blade height, b_{4} (m) | 0.0036 | 0.0039 | 0.0041 |

Inlet absolute flow angle, α_{4} (°) | 76.8186 | 76.8186 | 76.8186 |

Inlet relative flow angle, β_{4} (°) | −45.718 | −46.0327 | −40.3337 |

Inlet static temperature, T_{4} (°C) | 66.5 | 80.3 | 81.5 |

Inlet static pressure, p_{4} (kPa) | 394.8306 | 325.7557 | 371.049 |

Inlet absolute Mach number, Ma_{4} | 0.9591 | 0.8798 | 0.9586 |

Inlet relative Mach number, Ma_{4}′ | 0.2618 | 0.2392 | 0.2541 |

Outlet hub radius, r_{6h} (m) | 0.0222 | 0.0231 | 0.0207 |

Outlet shroud radius, r_{6s} (m) | 0.037 | 0.0376 | 0.0357 |

Outlet blade height, b_{6} (m) | 0.0148 | 0.0145 | 0.015 |

Outlet absolute flow angle, α_{6} (°) | 0 | 0 | 0 |

Outlet relative flow angle, β_{6_rms} (°) | −66.8922 | −66.6462 | −67.2171 |

Outlet relative flow angle, β_{6_h} (°) | −61.1328 | −61.1328 | −61.1328 |

Outlet relative flow angle, β_{6_t} (°) | −71.7233 | −71.3152 | −72.2554 |

Outlet static temperature, T_{6} (°C) | 37.5 | 47.4 | 53.5 |

Outlet static pressure, p_{6} (kPa) | 116.4663 | 107.3252 | 131.6736 |

Outlet absolute Mach number, Ma_{6} | 0.2357 | 0.2219 | 0.2299 |

Outlet relative Mach number, Ma_{6}′ | 0.5967 | 0.5617 | 0.5819 |

Number of blades, Z_{R} (−) | 18 | 18 | 18 |

Nozzle | |||

Inlet radius, r_{1} (m) | 0.0739 | 0.0769 | 0.0691 |

Inlet blade height, b_{1} (m) | 0.0036 | 0.0039 | 0.0041 |

Inlet static temperature, T_{1} (°C) | 79.9 | 94.4 | 96.7 |

Inlet static pressure, p_{1} (kPa) | 659.1412 | 505.5017 | 620.1825 |

Inlet absolute Mach number, Ma1 | 0.1955 | 0.1796 | 0.1891 |

Inlet absolute flow angle, α_{1} (°) | 49.1958 | 49.3194 | 49.4424 |

Outlet radius, r_{2} (m) | 0.0575 | 0.0599 | 0.0539 |

Outlet blade height, b_{2} (m) | 0.0026 | 0.0039 | 0.0041 |

Outlet absolute flow angle, α_{2} (°) | 76.8186 | 76.8186 | 76.8186 |

Outlet static temperature, T_{2} (°C) | 66.5 | 80.3 | 81.5 |

Outlet static pressure, p_{2} (kPa) | 394.8306 | 325.7557 | 371.049 |

Outlet absolute Mach number, Ma_{2} | 0.9591 | 0.8798 | 0.9586 |

Blade thickness (mm) | 0.59171 | 0.58982 | 0.57152 |

Number of blades, Z_{B} (−) | 25 | 25 | 25 |

Working Fluid | Number of Elements | ${\mathsf{\eta}}_{\mathit{t}}$ (%) | ${\mathit{P}}_{\mathit{t}}$ (kW) | $\mathbf{Difference}\text{}(\%)\text{}{\mathsf{\eta}}_{\mathit{t}}$ | $\mathbf{Difference}\text{}(\%)\text{}{\mathit{P}}_{\mathit{t}}$ |
---|---|---|---|---|---|

R-245fa | 413,845 | 76.12 | 8.76 | - | - |

629,391 | 76.73 | 8.71 | 0.8 | 0.57 | |

R-141b | 510,330 | 80.30 | 9.41 | - | - |

782,506 | 80.47 | 9.37 | 0.21 | 0.42 | |

R-123 | 418,276 | 77.60 | 7.37 | - | - |

627,414 | 78.20 | 7.40 | 0.77 | 0.40 |

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## Share and Cite

**MDPI and ACS Style**

Carrillo Caballero, G.; Cardenas Escorcia, Y.; Venturini, O.J.; Silva Lora, E.E.; Alviz Meza, A.; Mendoza Castellanos, L.S.
Unidimensional and 3D Analyses of a Radial Inflow Turbine for an Organic Rankine Cycle under Design and Off-Design Conditions. *Energies* **2023**, *16*, 3383.
https://doi.org/10.3390/en16083383

**AMA Style**

Carrillo Caballero G, Cardenas Escorcia Y, Venturini OJ, Silva Lora EE, Alviz Meza A, Mendoza Castellanos LS.
Unidimensional and 3D Analyses of a Radial Inflow Turbine for an Organic Rankine Cycle under Design and Off-Design Conditions. *Energies*. 2023; 16(8):3383.
https://doi.org/10.3390/en16083383

**Chicago/Turabian Style**

Carrillo Caballero, Gaylord, Yulineth Cardenas Escorcia, Osvaldo José Venturini, Electo Eduardo Silva Lora, Anibal Alviz Meza, and Luis Sebastián Mendoza Castellanos.
2023. "Unidimensional and 3D Analyses of a Radial Inflow Turbine for an Organic Rankine Cycle under Design and Off-Design Conditions" *Energies* 16, no. 8: 3383.
https://doi.org/10.3390/en16083383