## 1 Introduction  
The growing global demand for energy [1; 2], coupled with the imperative to reduce greenhouse gas emissions [3; 4], has intensified the search for efficient and sustainable energy conversion technologies [5; 6]. Waste heat recovery systems have emerged as a promising solution to enhance energy efficiency in various industrial processes by converting low-grade thermal energy into useful power [7]. Among these systems, the Organic Rankine Cycle (ORC) has gained significant attention due to its ability to utilize heat sources at relatively low temperatures, which are often inadequate for conventional steam-based power cycles [8; 9]. The ORC employs organic working fluids with favorable thermodynamic properties, enabling efficient energy conversion from low-temperature heat sources such as industrial waste heat [10], geothermal energy [11; 12], and biomass [13; 14]. The selection and optimization of key components within the ORC, particularly the expander or turbine, are of critical importance for maximizing system performance [15]. Axial [16], radial inflow [17], and radial turbines [18] are often preferred in small-to-medium-scale ORC applications due to their compact size, simplicity, and ability to handle significant volume flow rates and pressure ratios. While turbines offer numerous advantages [19], their performance in ORC systems is highly sensitive to operating conditions, particularly when exposed to variable waste heat sources [20; 21]. Industrial processes frequently exhibit fluctuations in waste heat availability due to changes in production rates, process interruptions, or seasonal variations. Exceeding the design point of a turbine can result in inefficiencies [22; 23], mechanical stresses [24], and a reduction in lifespan [25]. Consequently, it is crucial to comprehend the behavior of turbines under variable operating conditions to optimize ORC systems and ensure reliable and efficient energy recovery. The current research in this field has concentrated on the design and optimization of radial turbines for specific operational conditions [26; 27]. In some studies, multiple operational points have been taken into account [28]. In article [29], the authors adopt a multipoint design method to enhance turbine performance under complex operating conditions. Their findings demonstrate that implementing this multipoint optimization approach effectively eliminates flow separation across a spectrum of operating conditions, thereby ensuring optimal turbine performance even when operating outside the designed parameters. Nevertheless, these considerations have not yet permitted the construction of a loss model that would enable the prediction of turbine operating parameters over a broad range of operating characteristics at the design stage. However, there is a paucity of research addressing the challenges associated with turbine operation under fluctuating waste heat conditions. The conventional approach often involves designing turbines for a fixed operating point, which may not be optimal when the heat source varies. This gap necessitates a systematic investigation into turbine adaptability and performance across a range of operating conditions, with the aim of developing designs that can maintain high efficiency despite variations in waste heat input. The objective of this study is to numerically investigate the performance of a multi-stage radial turbine under variable waste heat conditions relevant to ORC applications. By employing advanced CFD techniques, the research aims to analyze the turbine's behavior beyond its design point, identify performance limitations, and propose solutions to enhance adaptability and efficiency. The significance of this work lies in its potential to inform the design of turbines capable of operating effectively across a spectrum of conditions, thereby maximizing energy recovery from variable waste heat sources.