1. Introduction
The global energy crisis poses a significant challenge to sustainable development [
1]. Fossil fuels are declining due to fluctuating prices and reserves, as well as geopolitical and environmental issues caused by their rapid consumption. To address this challenge, low- and medium-temperature heat sources, such as renewables (solar, geothermal, biomass, and ocean thermal energy) and wasted heat from various industrial processes (such as petrochemical plants, gas turbines, and internal combustion engines), are being widely used [
2].
ORC systems, which use organic working fluids, are a flexible and efficient solution for converting low-grade heat sources, such as biomass, waste heat, and solar energy, into electricity [
3]. Many types of organic fluids can be used as working fluids in ORC systems, including hydrocarbons, hydrofluorocarbons, hydrochlorofluorocarbons, chlorofluorocarbons, perfluorocarbons, siloxanes, alcohols, aldehydes, ethers amines, and inorganic fluids. ORC systems are widely applicable, secure, and environmentally friendly, making them well-suited to low- and medium-temperature energy conversion technologies [
4]. ORC systems could be of significant benefit, both environmentally and economically [
5]. However, some components of ORC systems, particularly turbines, can be expensive to design, which is impractical for small-power-load ORC systems that cannot bear the cost of each component’s design. Thus, it is essential to consider the off-design performance of the same component to optimize the system’s overall cost-effectiveness. Similar to the traditional steam Rankine cycle, the ORC expands high-pressure working mediums to low pressure for power output. Also, the components are the same, which include a generator, expander, condenser, pump, etc. However, the main difference between them is the working fluid: the ORC uses an organic compound with an evaporation temperature lower than that of water, which allows for power generation from low-and medium-temperature heat sources [
6].
The ORC system is a unique power cycle that is suitable for low-temperature heat sources, and the same equipment can be used with different types of working fluids [
7]. ORC applications can be found in a wide range of power outputs, including micro/mini/small/medium/large ORCs (<5 kW, 5–50 kW, 50–500 kW, 500 kW–5 MW, and >5 MW) [
8]. The turbine is the primary component responsible for converting heat into power in ORC systems and can be classified into various types, such as axial flow, centripetal, piston, scroll, and screw turbines [
9]. Researchers have been focusing on optimizing the performance of the turbine and increasing the efficiency of ORC systems. For example, Zhao et al. [
9] reviewed various expansion devices for ORC systems using low-temperature heat recovery, while Kolasinski [
10] proposed a method for selecting the most suitable working fluid for ORC systems employing volumetric expanders. Zhang et al. [
11] investigated the selection of zeotropic mixtures for ORCs using single-screw expanders. Hou et al. [
12] employed the supercritical carbon dioxide cycle and organic Rankine cycle using a mixture of cyclopentane/R365mfc, resulting in high system efficiency and low costs.
Feng et al. [
13] found that the use of toluene in dual-source ORC systems can significantly improve fuel economy and reduce carbon dioxide emissions. Additionally, studies have significantly contributed to the development of ORC technology and the optimization of turbine performance [
14,
15,
16,
17,
18]. While the cost of design for some ORC components can be high, especially for turbines, it is meaningful to consider the off-design performance of the same component for small-power-load ORC systems to bear the design cost of each part.
In the field of ORC turbine research, much attention has been focused on maximizing efficiency through mainline models and predicting turbine performance under off-design conditions. However, many papers propose the design of a new turbine for each specific working condition, which can be costly, especially for small-scale turbines. To address this issue, Fiaschi et al. [
19] proposed a method that couples fluid selection and turbine efficiency estimation methods to improve the basic design of the turbine. They also proposed a 0D model to predict the performance of radial turbines for ORCs. Jubori et al. [
20] compared the performance of both axial and radial turbines in ORCs and found that the axial turbine performed better. Sauret and Gu [
21] used CFD simulations to analyze the performance of a turbine with R143a as the working medium, demonstrating that the efficiency changed significantly with large variations in the fluid. White et al. [
22] used similitude theory to adapt an axial turbine to work efficiently with different fluids, improving the economy of scale. CFD simulations have been widely used in many studies to explain the laws of parameter changes, as it is both conventional and effective. For example, Mohammad et al. [
23] used the CFD method to evaluate the performance of the micro wind turbine, Michael et al. [
24] identified optimal operating conditions based on a given turbine and real gas as the working medium, while Mario et al. [
25] used 3D CFD methods and 2D PD calculations to verify the performance of a small multi-stage axial turbine. Dawid et al. [
26] optimized ORC turbine design using CFD technology to improve power generation efficiency.
Based on the aforementioned survey, it is apparent that there is a lack of off-design analyses of axial turbines in ORC systems. However, the axial flow turbine in the ORC system requires off-design analysis to assess its adaptability to varying operating conditions and output requirements. This analysis is crucial for reducing design costs and improving the system’s renewable energy utilization efficiency.
To achieve this goal, a 3D model of the turbine was built using physical data. CFD simulations were then used to analyze the turbine’s performance, and the results were analyzed to discuss the characteristics of the turbine and its influencing factors. Moreover, the reliability of the simulation model was verified by the experiment. The contribution of this paper is the analysis of a small-scale ORC turbine to make it more suitable and reduce design costs for different available ORC systems. At present, most of the research on ORC turbines is focused on centripetal turbines. This article mainly studies axial flow turbines and verifies their good off-design characteristics.
3. CFD Simulations
3.1. Model
The 3D model of the fluid domain and the turbine’s flow passage are shown in
Figure 1b, with an average diameter of the vanes and blades of 77 mm. The turbine has 6 vanes for 0.5 partial admissions and 27 blades, with both blades and vanes having a height of 5 mm. Under the design parameters, the turbine mass flow rate is 0.3 kg/s, and the power output of the turbine is 5 kw. The fluid enters from the upper end-face in
Figure 1b, while the outlet section’s flow area is extended to promote full liquidity development.
3.2. Mesh
The structured meshes were generated by the multizone method in ANSYS meshing based on the 3D model, with each vane having approximately 32,500 meshes and each blade having approximately 22,000 meshes, and the blade surface boundary was set with a boundary layer. The entire model has around 850,000 hexahedral meshes, with a minimum orthogonality mesh angle of 38.5 degrees. This minimum angle is much larger than that required by conventional computational fluid dynamics calculations, ensuring the stability and reliability of the simulation results. From
Table 1, it can be seen that as the number of grids increases from 425,000 to 850,000, the temperature of the working fluid at the outlet of the stationary blade changes significantly. However, as the number of grids increases from 850,000 to 3,400,000, the temperature of the working fluid at the outlet of the stationary blade remains basically unchanged, which shows the reliability of the mesh.
3.3. Numerical Method
Various conservation equations are being followed by the flow in the turbine, including the continuity equation, momentum equation, energy equation, turbulent kinetic energy and turbulent dissipation-rate equation, and k-ε equation. The k-epsilon turbulence model is the most common turbulence model; it boasts high computational efficiency, extensive applicability, and is capable of accurately delineating the flow intricacies of the working fluid within the turbine, while simultaneously providing a comprehensive reflection of the turbine’s overall performance. This belongs to the two-equations model and is suitable for fully developed turbulence. By coupling the transport equations of k and ε, as well as the momentum equation of average velocity, the continuity equation, the thermal equilibrium equation, etc., both k and ε can be solved.
The conservation equation is expressed with the following equation:
where Φ is the variable in the conservation equation, Γ is the corresponding generalized diffusion coefficient, and S is the source term.
For the continuity equation, there is no diffusion term and source term:
For the energy conservation equation,
For the momentum equation,
x-direction
y-direction
z-direction
ANSYS CFX 2022 software is used for numerical calculations. The finite volume method is used for discrete conservation equations, and the SIMPLE algorithm is applied to simulate the flow field. The purpose is to simulate the flow by the fluid on the turbine in the impeller passage.
3.4. Boundary Conditions
The previous section outlined the design condition and operating conditions of the ORC turbine model. In this section, the off-design conditions are presented. The off-design conditions are listed in
Table 2, which includes the total inlet pressure, rotor speed, and inlet total temperature. The total inlet pressure is raised from 0.65 MPa to 0.85 MPa at 0.05 MPa intervals, and the rotor speed is increased from 17,500 rpm to 18,500 rpm at 500 rpm intervals. A superheat is set in the off-design conditions to prevent the medium from undergoing a phase transition due to the different total pressure.
It is important to analyze the off-design conditions of the turbine as it reflects the adaptability of the turbine to different operating conditions and output requirements, which is crucial for designing wide-operating-condition turbines. The off-design analysis can also help in saving the design cost of a turbine for different systems.
3.5. Main Performance Indicators
In order to analyze and evaluate the performance index of the turbine, some main indicators should be acquired through the post-processing of simulation results.
Power output:
where ω is the rotor speed (rads/min) and
T is the torque of the blades (Nm).
The isentropic efficiency:
where m is the mass flow (kg/s) and Δh is the theoretical isentropic enthalpy drop from the inlet to the outlet of the medium (kJ/kg).
The total pressure recovery:
where P
t is the total pressure and P
s is the static pressure.
The velocity triangle shows how the medium works in the turbine. To establish this, we must obtain the mass flow average circumferential and radial velocities at the vane outlet and blade outlet from the numerical calculation.
The load factor can be obtained from the velocity triangle:
where ΔC
u is the circumferential velocity change and U is the implicated velocity, which depends on the rotor speed.
3.6. Validation
To ensure the accuracy of the numerical calculation, a verification process was conducted by comparing the simulation results with experimental data. The experimental apparatus was obtained from a small-scale ORC experiment system described in reference [
27].