Design and off-design optimisation of an organic Rankine cycle (ORC) system with an integrated radial turbine model
Introduction
The production of electricity from fossil fuels, renewable sources, waste heat or even the storage of electricity itself requires careful attention to the underlying thermodynamic cycles if these energy sources are to be exploited efficiently and cost-effectively [13], [14]. There are significant differences between these energy sources in terms of their magnitude, temporal variation and (in the case of thermal sources) their temperature. Consequently, different thermodynamic cycles are appropriate for their exploitation. For example, ORCs are deemed particularly suitable for low-to-medium temperature and small-to-medium scale waste heat recovery [31], [32], [7], [30], whilst a range of solutions has been proposed for thermo-mechanical energy storage systems, including supercritical/transcritical CO2 cycles [37], Joule-Brayton based pumped thermal storage [9], [68], [36] and many others. In order to select and to design the best cycle and choose the most appropriate working fluid, the individual cycle components (heat exchangers [24], [10], compressors, expanders [75], [65], [21], etc.) need to be modelled. Early-stage cycle design can be carried out with simple modelling methods involving relatively few parameters, such as an effectiveness for heat exchangers or a single isentropic (or polytropic) efficiency for compressors and expanders. Ultimately, however, more detailed information is required, and it is often necessary to know how a given component will perform over a range of operating conditions, away from its nominal design point. Thus, a detailed and reliable system analysis requires also in-depth understanding of the component models. Therefore, this paper contributes to the ORC system analysis by establishing and validating a component model for radial turbines as well as exploring the technical possibilities for part-load operation of the component and the system.
In the early stages of system design, the radial turbine is often incorported into the system calculation with an estimated peak efficiency, obtained by choosing two size parameters – for example the loading coefficient and the flow coefficient. For this purpose, Chen & Baines [5] correlated the total-to-static efficiency of around 40 different radial turbines (most of them if not all operated with fluids behaving like ideal gases) with the loading and flow coefficients, thereby establishing that maximum efficiency can be reached for loading coefficients between 0.9 and 1.0 and flow coefficients between 0.2 and 0.3. The mean value of total-to-static efficiency in these ranges is around 87%. Similar maps have been compiled for organic fluids [27], [29]. White & Sayma [73] concluded that their data for different working fluids correlates well with the data of Chen & Baines [5]. White & Sayma [74] presented a method that predicts the turbine design point efficiency only from the thermodynamic conditions and applied this model to cycle optimization, obtaining different optimum cycles with the new method compared to a fixed isentropic efficiency. The possibility of generalising turbine performance data with reduced performance maps and similitude theory was studied extensively for ORC fluids by White & Sayma [69].
The aforementioned methods (size parameters, reduced performance maps, similitude theory) are a reliable and robust way to incorporate the turbine efficiency into thermodynamic cycle calculations and into the ORC system analysis if only the design point is considered, the number of design options is small, or the number of fluids is limited. Clearly a greater level of detail is required in order to model off-design behaviour and/or a larger number of fluids, that can even be designed during the cycle optimisation [43], [72], [60].
The aim of the present paper is to provide a validated calculation approach for radial turbines suitable for incorporation into cycle calculations with the intention to improve the ORC system analysis. (Note that the focus is on thermodynamic and aerodynamic performance – mechanical losses, electrical losses and costs are not considered here.) The paper starts with a review of the relevant experimental work and literature that gives comparison of experimental data with computations. The mean-line approach is then outlined, followed by validation against experimental data and supported by CFD calculations. Finally, the validated mean-line code is applied in the context of an ORC, considering both design and part-load behaviour to help exploring the technical possibilities offered by a radial turbine for on- and off-design operation of a whole ORC system.
The history of radial turbines spans several decades and for most purposes the technology is well understood. The associated literature is extensive, ranging from design approaches, mean-line and CFD analysis, to experimental verification. However, there are still phenomena, such as condensation, that limit the operational range and that are not fully understood (see [52]), and studies on the impact of real gas effects are currently ongoing (e.g., [66], [67], [12], [71]). Given the wide scope of this literature, the purpose of the present section is to give a broad but comprehensive overview of the work most relevant to mean-line modelling and to highlight the main findings.
The main experimental contributions come from Ricardo and Co., the NASA program, the Politecnico di Torino, Sundstrand and the Belfast program. Not all of these, however, provide sufficient information for the current purposes. Certain geometric information must be available to validate calculation methods. For mean-line calculations, the basic requirements for modelling isentropic changes include information on the nozzle inlet, tip and outlet diameter, the nozzle height and blade height, and the blade or flow angles. To model losses, the nozzle vane chord length, number of nozzle vanes, number of rotor blades, and clearances are all needed in addition. With this in mind, the following points apply to the above-mentioned list:
- (i)
The reports from Ricardo & Co. are not available in the open literature but a summary is given by Hiett & Johnston [20]. The geometry of the rotor is not fully described and the outlet blade angle is only given on an arbitrary diameter. Thus, the geometry can only be reproduced with certain assumptions.
- (ii)
NASA tested two radial turbines with tip diameters of 12 cm and 15 cm in the 1960s and early 1970s [26], [64], [41], [63], [16], [17]. The impact of rotational speed, Reynolds-number, tip clearance height and the use of splitter blades, mainly on the total-to-static efficiency, can be assessed from the NASA technical notes. An important finding from Nusbaum & Wasserbauer [41] is that the peak total-to-static efficiency increases (by 2.1 percentage points) when the Reynolds number () changes from 64 000 to 352 000. This efficiency change is rather small compared to the efficiency spectrum over a whole operational range.
Diameters and blade heights are fully described in the early NASA reports although some values must be derived. E.g. the nozzle height must be derived from the absolute and the relative rotor axial tip clearance. The design velocity triangles are given but nothing is said about the blade angles. Thus, to use the early NASA data for code validation assumptions must be made on either the blade angle or the flow angle (e.g., flow angles are equal to design flow angles).
- (iii)
In the later NASA publications [35], [55] a full description of the radial turbine geometry is provided. Therefore, these cases can be used for mean-line model validation as well as for CFD code validation. The first publication is more promising since the whole operational range is investigated instead of a single point.
- (iv)
The Politecnico di Torino paper [8] is missing information on the nozzle inlet. Thus, the nozzle losses cannot be calculated. Also, no information on the rotor clearance is given, so the results are not suitable for validation.
- (v)
In the Sunstrand paper [22], measured values are given for the total-to-static and the total-to-total efficiency over a range of pressure ratios and rotational speeds while the mass flow rate is only given for one operation point. This is a drawback compared to the NASA investigations.
- (vi)
Plots of measured efficiency in the Belfast program [56], [3], [57], [11], [58] show qualitatively different behaviour from the above-mentioned measurement campaigns. Since no information is disclosed to fully reproduce the geometry, an in-depth analysis for this difference is not possible and so these results are not considered further.
- (vii)
Recently, the growing interest in ORC technology has pushed experimental investigations on radial turbines working with ORC fluids forward, and Kang [23] found maximum turbine efficiencies above 80% for a turbine working with R245fa, while Shao et al. [53] reported an efficiency value of 85% for a turbine working with R123a. These developments suggest that in the near future experiments on ORC turbines with fully disclosed geometries will be available for further validation.
Besides the experimental investigations, mean-line and CFD program developments have a direct impact on this paper. For instance, Wasserbauer & Glassman [62] described a mean-line program that requires the nozzle and rotor loss coefficients at the design point to be specified. Together with certain geometric values the off-design performance can be computed. These are in excellent agreement with the experimental results of Nusbaum & Kofskey [40] and Wasserbauer et al. [64].
Simonyi & Boyle [54] compared quasi-3D calculation results with experiment. The computed total-total efficiency is four percentage points higher than the measured values. The slope of the efficiency curve plotted as a function of the pressure ratio also shows a considerable discrepancy, which they trace back to an uncertainty in the incidence loss model.
Baines [4] compared the Ricardo & Co. results with a mean-line model and achieved excellent agreement. In a later publication [45] mean-line results are compared to experimental results from Ricardo & Co. and NASA and although the agreement is mostly excellent, the authors state that some tuning of the modelling parameters is required to achieve this. In particular, agreement with the C80 case was poor and required a very large increase in the rotor passage loss to match the measured efficiency [45]. Persky & Sauret [44] compare 1.5 million loss model configurations applied to a radial turbine working with either R134a or CO2 and concluded that only a few are suitable to predict the turbine efficiency at off-design conditions.
The Jones data are widely used for CFD code validation since Sauret [49] rediscovered this case and presented CFD results. Several authors [49], [25], [1] predicted higher efficiencies of between two and four percentage points. Odabaee et al. [42] however, calculated a considerably smaller efficiency by about 5–6 percentage points whilst Gad-el Hak et al. [18] got a nearly perfect match (within 0.3 percentage points). These differences are at first surprising, given that they were obtained with the same CFD code, but most likely stem from uncertainties in specification of the geometry. Only Anderson & Bonhaus [2] compared CFD results to the experiments for the Ricardo & Co. R70B turbine. Excellent agreement was obtained but, again, nothing is said about how the geometry was reproduced. Finally, Wheeler & Ong [67] compared CFD results to the experimental results obtained by McLallin & Haas [35] for the fully-described turbine and predicted a total-to-static efficiency just 0.5 percentage points lower.
Mean-line as well as CFD methods are applied to ORC turbines in the literature. Rahbar et al. [46] show the difference in efficiency for different turbines. The present paper continues this by also considering two-stage turbines to overcome efficiency penalties at high pressure ratios/loading coefficients and extends the investigations to off-design operation. Integrated design programs are often checked for plausibility in terms of computed geomerty and efficiency against experimental results and other design programs [61], [47]. Thereby, Rahbar et al. [47] obtained a ratio of hub radii and an efficiency difference of 3 percentage points. In the present paper, the turbine design process is separated from the performance calculation, this eases the validation process since the experimental investigated geometry can be modelled directly in the mean-line program. Further, the validation can simply be extended to off-design conditions.
In order to optimise a turbine, different levels of detail are considered in the open literature: Mean-line models can only be used to optimise the main geometry dimensions, e.g., diameters and the desired flow angles at inlet and outlet. A further optimisation can be done by 3D CFD as shown by Sauret & Gu [50]. White & Sayma [70] presents a method to predict the fluid velocity distribution inside the rotor passage with a reduced order model. Zheng et al. [76] combine a design tool with CFD as well as with a mean-line off-design tool and general trends of turbine power and efficiency with changing rotational speed, pressure ratio, and turbine inlet temperature are shown. In continuation to the aforementioned studies, this paper extends the existing knowledge by applying and optimising adjustable nozzle guide vanes to maximise cycle efficiency at off-design operation.
The above literature act to motivate the present work where point iii) and iv) are the main contributions:
- (i)
Comparison and validation of a unified mean-line model for radial turbines against a range of experimental and CFD results in terms of pressure ratio and total-static isentropic efficiency including off-design operation considered with mass flow rate and rotational speed variation.
- (ii)
Integration of a validated mean-line model into an ORC system model that enables explorations of the off-design operation and performance of sub-critical and transcritical systems.
- (iii)
Extension of the suitable design point pressure ratio through the employment of multi-stage radial turbines.
- (iv)
Operation range extension of ORC systems by utilizing adjustable guide vanes.
Section snippets
Description of the main features of radial turbines
A radial inflow turbine consists of nozzle vanes and a rotor, as depicted in Fig. 1. The left-hand figure shows the so-called meridional plot (a cut between the blades) and that on the right shows the blade-to-blade view (at roughly mid blade height) and provides an indication of the flow turning. This is important since all turbomachinery operates by changing angular momentum and hence turning the flow in the circumferential direction. This is further highlighted by the velocity diagrams drawn
Application to non-regenerative ORC system
The validated mean-line model is now applied to a non-regenerative ORC system to investigate the most suitable pressure ratio range and the impact of part-load operation on the cycle efficiency. Since the focus of this work is on the radial turbine, other components are modelled using very simple approaches. For example, the turbine inlet temperature is assumed fixed at 423 K (i.e., the dynamics of the main heat exchanger are not considered), but a 5% pressure loss is assumed in both the heat
Conclusions
Motivated by the growing interest in ORC system design this paper presented a mean-line method for radial turbines with a specific view towards its application to thermodynamic cycle analysis. This has included a review of data and loss correlations available in the literature, which have subsequently been used to establish the mean-line model with a geometric description appropriate to radial machines. The model has been validated against a range of published experimental results for on- and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) [Grant No. EP/P004709/1]. Data supporting this publication can be obtained on request from [email protected].
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