Design guidelines for supersonic stators operating with fluids made of complex molecules

Supersonic stator vanes account for two-thirds of the ﬂ uid-dynamic losses in high temperature mini- Organic Rankine Cycle (ORC) turbines. As a result, the overall performance of the turbo-expander mainly depends on the design of the stator. Currently, there is no established correlation for the optimal design of such cascades. This work concerns an investigation about the accuracy of the only design method currently available for the design of supersonic stators operating with ﬂ uids made of complex molecules. A physics-based analytical model and a CFD-based model were developed to estimate the optimal post-expansion ratio and to compare their results with the Deych ’ s model. The analysis shows that the Deych ’ s method fails to accurately predict the optimum value of the post-expansion ratio. The study covers also the assessment of the optimum post-expansion ratio in relation to the solidity, the design ﬂ ow angle and the total-to-static expansion ratio. The outcome demonstrates that there exists a unique optimum post-expansion ratio for a set of primary stator design parameters. In summary, vanes operating with a substance made of complex molecule as the working ﬂ uid, which is typical of high-temperature ORC turbines, feature a unique theoretical value of the optimum post-expansion ratio for a given total-to-static expansion ratio. New correlations are required to predict this value.


Introduction
The high-expansion ratio is a defining feature of unconventional turbomachinery, i.e., turbomachinery operating with nonconventional working fluids like those treated for example in Ref. [1,2]. Given that such working fluids are made of complex organic molecules for which the speed of sound is quite low, the stator often operates at highly supersonic flow conditions [3,4]. As a result, new and ad hoc fluid dynamic design methods have been developed in recent times.
ORC power systems are becoming increasingly relevant because they can efficiently convert thermal energy from renewable and waste heat sources [3]. Regardless of its extremely wide application range, the full potential of ORC technology is not realized yet. Economic viability will increase if current technical challenges affecting system performance versus cost are overcome. One issue affecting turbine design and performance of high-temperature high-expansion ratio machine is the complex and non-ideal fluid dynamics within the stator [5]. It has been recently shown, see Refs. [6], that the fluid dynamic losses occurring in the stator of an existing high-temperature high-expansion ratio ORC turbine amount to approximately two-thirds of the total losses of the turbine. This is the case in such type of small-capacity turbines, which are always highly loaded in order to limit the number of stages to unity because of economic constraints. Therefore, the design of efficient stator cascades is critical.
The preliminary design of supersonic stationary vanes involves the choice of several parameters, e.g., the flow angle (4 a ), the expansion ratio (b t1 ), the degree of divergence (DoD), the solidity (s) etc., based on correlations. The blade profile is then obtained with the Method of Characteristics (MoC) and refined by means of computational fluid dynamics (CFD) simulations [1]. The mechanical verification of the fluid dynamic design completes the design iteration. Recent studies already dealt with the detailed fluid dynamic design of ORC stators [7,8]. However, the optimal choice of preliminary design parameters dictating the subsequent detailed fluid dynamic design phase has not been treated in the literature yet. Currently, therefore, correlations which were developed to obtain preliminary design parameters for conventional gas turbines are adopted, and the optimal design of supersonic ORC blades can only be obtained by resorting to computationally expensive numerical optimization based on high-fidelity CFD, and starting from highly suboptimal configurations.
The degree of divergence is the preliminary design parameter that has the largest impact on the resulting fluid dynamic performance of supersonic ORC vanes [8]. To the authors knowledge, the only model that can be currently employed to calculate the optimal degree of divergence is the one proposed by Deych [9]. This model provides an estimate of the optimum degree of nozzle divergence (DoD) as a function of the stator exit Mach number. The model was derived for the case of supersonic axial cascades operating with air as working fluid, therefore its application to the design of ORC stators affected by non-ideal compressible flow effects is arguably unjustifiable from a theoretical point of view. The objectives of the study documented in this paper are (i) to discuss the accuracy and limitations of the Deych's model, henceforth referred to as empirical method, if applied to nozzles operating with fluids different than air, and (ii) to assess the variability of the optimal post-expansion ratio with respect to primary design variables, namely, the flow metal angle, the solidity and the expansion ratio.
To achieve these goals, a physics-based and a CFD-based loss estimation models, limited to perfect gas flows, were developed and these are referred to as analytical method and CFD-based method in the following. The analytical method estimates the boundary-layer and mixing losses for a typical supersonic axial vane configuration. The losses in the boundary-layer are evaluated according to the procedure described in Ref. [10]. The mixing loss is evaluated by applying the mass, momentum and energy conservation equations to the semi-bladed region of the stator vane, as explained in Refs. [11]. The CFD-based method comprises of a detail design procedure and a simulation. The vane geometry generation method is adapted to handle axial configurations from the method described in Ref. [8]. The fluid-dynamic performance of the blades is calculated using the open-source Reynolds averaged Navier Stokes (RANS) solver SU2 [12]. The flow equations are complemented by the one-equation Spalart-Allmaras turbulence model [13].
In order to investigate the accuracy of the empirical model, the optimum DoD (or alternatively, the optimum post-expansion ratio) obtained from this model is compared to the values calculated with the numerical and the analytical methods. Additionally, the analysis is extended to consider working fluids ranging from those made of simple molecules, to those made of complex organic molecules in order to study the effect of the fluid molecular complexity on the optimum post-expansion ratio. The variation of the optimum post-expansion ratio with the primary design variables is assessed by scanning the design space with the CFD-based method.

Methodology
The flow domain of a typical axial stator vane can be divided into three sections, see Fig. 1 (a): the converging section between station (t) and (o), the diverging section between station (o) and (a) and the semi-bladed section between station (a) and (1). The assessment of the empirical method is based on the comparison of the optimal DoD calculated with the empirical method with those obtained with the analytical and the CFD-based methods.

Empirical method
In 1965, Deych illustrated a method to estimate the preliminary DoD for a stator vane operating in transonic/supersonic flow conditions [9]. To date, this is the only method available and adopted in common practice. According to this method, the optimum DoD is defined by the relation where, M 1 is the stator design Mach number, A o is nozzle throat width, A a is nozzle outlet width, A is is the area ratio corresponding to an isentropic expansion and g is the heat capacity ratio. The current design practice is thus that nozzles for which the design M 1 is lower than 1.4 (transonic) are made without the diverging section, because the semi-bladed section provides the flow with a sufficient degree of expansion without an excessive penalty in terms of efficiency. In contrast, supersonic nozzles (M 1 > 1:4Þ are always designed with a diverging section.

Analytical method
Two of the major dissipation mechanisms in supersonic axial vanes are due to mixing and viscosity in the boundary layer. The procedure to analytically estimate these losses is as follows: (a) Mixing losses. The amount of kinetic energy that is dissipated due to mixing is estimated, as described in Ref. [11], by solving the mass, momentum and energy conservation equations in the control volume encompassing the semi-bladed section of the vane, see Fig. 1(a). The flow deviation angle is D4 ¼ arctan where, M a is the nozzle Mach number, 4 a is the design flow angle 1a ≡b a1 is the post-expansion ratio. The only independent quantity in Eqn.
(2) is b 1a which allows to evaluate M 1 at the exit of the vane. Thus, the amount of kinetic energy that is dissipated due to mixing can be estimated as where, h is enthalpy, and subscript "is" stands for isentropic.
(b) Boundary layer losses. These losses are due to viscous dissipation within the nozzle boundary layer. Assuming turbulent flow on the blade surface, the kinetic energy dissipation can be calculated as where, In these equations, C d is the dissipation coefficient, r is the density, T is the temperature, S is the entropy, _ m is the mass flow rate, v is the velocity, c is the speed of sound and subscript "xt" stands for isentropic ratio between properties at station "x" and "t". C d is equal to 0.002 according to Refs. [10].
The only unknown in Eqn. (5) is the property distribution along the surface of the stator, for which the quasi-1D flow approximation is assumed valid. Since the boundary layer losses in the subsonic section are significantly lower than the boundary layer losses in the supersonic section (as boundary layer losses scale with the cube of the Mach number), only the diverging part of the nozzle and the semi-bladed section are modeled for simplicity. In the diverging section of the nozzle the flow accelerates, while in the semi-bladed section the linear suction side surface guides the flow to the outlet, see Fig. 1(a). The Mach number distribution is computed according to the 1D flow approximation by assuming a linear variation from unity at the throat to M a at the nozzle outlet section, see Fig. 1(b). The suction side is modeled as a flat plate over which the fluid flows at constant Mach number M a , see Fig. 1 Eqn. (5) applied to the geometry depicted in Fig. 1(b) allows to compute the boundary layer loss in the supersonic vane as where, x bl j L 0 is the loss due to the nozzle surface and x bl j Lte L is the loss due to the linear suction side.

CFD-based method
The vane geometries with varying DoD are generated and analysed by following a procedure which can be divided into two parts, namely, geometry generation and CFD simulation. The procedure is similar to that documented in Ref. [8] for radial vanes, however, it differs regarding the way the axial vane geometry is obtained.

Geometry generation
In order to design an axial supersonic stator, the nozzle geometry obtained by applying the MoC [14,15] is adapted to the vane configuration and the geometry of the converging section is generated by means of Non-Uniform Rational Basis Splines (NURBs) [16]. Details of this MoC are provided also in Ref. [8], while the procedure to create the vane geometry starting from the nozzle obtained with the MoC is as follows.
1. Scaling factor. With reference to Fig. 2, two points ss te and ss * te , which are a pitch-distance (p) apart, are placed on the plane where the trailing edge of the stator is to end. Next, point ss a is obtained by the intersecting the line originating from point ss te at design flow angle 4 a and the line originating from ss * te at an angle 90 + À 4 a . The line ss a -ss te , see Fig. 2(a) can then be readily drawn. Point ps * a is positioned at a distance equal to the prescribed trailing edge thickness (t) from point ss * te along the line ss * te -ss a .
The diverging-nozzle geometry resulting from the application of the MoC is scaled such that the non dimensional width of the throat is unity, thus determining the value of the exit area of the diverging-nozzle A is as a function of M a . In order to complete the construction of the diverging part of the nozzle, the outlet crosssection width calculated with the MoC must be made the same as ss a -ps * a , see Fig. 2(c). This is accomplished by means of a scaling factor S determined by calculating the ratio of the distance between points ss a and ps * a and A is , namely as where, l is the length between ss a and ps * a .
2. Transformation of the nozzle geometry. After the nozzle geometry obtained with the MoC is scaled to the factor S , the scaled nozzle is then rotated by an angle 4 a about the center of the throat. The obtained geometry forms therefore the diverging section of the supersonic vane, see

CFD simulation
The performance of the supersonic vanes is computed with the aid of simulations performed with the open-source CFD software tool SU2 [12,17]. The computational domain encompassing one vane geometry is generated using UMG2, an in-house meshing program [18]. An exemplary geometry is depicted in Fig. 3. The computational domain is discretized using clustered quadrilateral elements close to the wall, ensuring a y þ < 1, and triangular elements in the rest of the flow domain. Following from a grid convergence study [19], a mesh of approximately 50,000 elements was chosen as the optimal compromise between accuracy and computational cost. The difference in the value of kinetic energy loss coefficient between the finest mesh and the one selected is 2%. The flow is simulated by solving the RANS equations and the turbulence equations are closed using the one-equation Spalart-Allmaras model [13]. The thermo-physical properties of the fluid are calculated using the polytropic perfect gas model.
The flow simulation is initialized by imposing a uniform state to the fluid everywhere in the domain, with properties corresponding to those of the total conditions at the inlet, and a value of the back pressure at the outlet. In addition, the non-reflective boundary condition, see Refs. [20], is also prescribed. The solution was obtained by using an Euler implicit time-marching scheme with a CFL of 10 while ensuring second-order spatial accuracy. Residual reduction of 4 orders of magnitude was achieved by running the simulations for a maximum of 4000 iterations.
The kinetic energy loss coefficient, accounting for both boundary-layer and mixing losses, is calculated numerically by obtaining mixed-out average quantities at the boundaries, resulting in where, h 1 is the averaged enthalpy at the outlet of the stator domain.

Case studies
In   In order to preliminarly assess the variation of b opt;a1 with the stator design parameters namely, solidity (s), design flow angle (4 a ) and total-to-static expansion ratio (b t1 ), vanes operating with toluene were selected. The fluid dynamic performance of the vanes was assessed by using the CFD-based method.

Results
Firstly, the accuracy of the empirical method is documented by means of a comparison of its results with the results of the analytical and the CFD-based methods for the considered working fluids. Next, the variation of b opt;a1 with design parameters is reported.

Comparison among the considered methods
The three methods, namely the empirical, the analytical and the CFD-based methods, are employed to calculate the optimum degree of divergence (or alternatively optimum post-expansion ratio) which ensures the least amount of fluid-dynamic losses for the considered test cases. Fig. 4(a) illustrates the results obtained with the analytical method, that is, the trends of boundary-layer losses, mixing losses and total losses as a function of b a1 . As expected, the mixing losses are minimal for b a1 ¼ 1 and the boundary-layer losses increase with the nozzle exit Mach number, i.e., with decreasing b a1 .
Moreover, the rate of change of the absolute value of the boundarylayer losses is much lower than that of mixing losses. Furthermore, both the boundary-layer and the mixing losses are larger for working fluids made of increasingly complex molecules, possibly because the flow Mach number is increasingly higher for the same operating conditions. The trend of the total loss x tot , see values, which is close to unity in all cases, as mixing-losses dominate the total loss in the stator.
The fluid-dynamic performance of the exemplary vanes are also calculated using the CFD-based method. The s of the vanes is maintained within 1±0:01 in accordance with the assumption made for the analytical method. Fig. 4 The cubic functional form was chosen somewhat arbitrarily after inspection of the trend of the total loss distribution computed with the analytical model. As opposed to the previous findings, it can be observed that the total loss is inversely proportional to the fluid molecular complexity, i.e., the total loss is higher for flows of fluids made of simpler fluid molecules. This is attributed to shock wave losses and the dissipation due to the impingement of the shock wave on the suction side of the blade. Nonetheless, the increase in total losses from b a1 ¼ b opt;a1 to b a1 sb opt;a1 is higher for fluids made of complex molecules. Finally, Fig. 4(b) shows also that the value of b opt;a1 is different depending on the working fluid, and it increases with decreasing molecular complexity.
Additionally, it can be observed that the values of x CFD is highly sensitive to small changes in the post-expansion ratio b a1 , and they become more scattered for b a1 away from unity. This is primarily due to the change in the position and strength of the shock wave emerging from the trailing edge of the stator vanes, which plays a key role regarding the adjustment of the flow field with respect to the imposed back pressure. Additionally, the interaction of these shock waves with wake and boundary layer influence substantially the overall performance.  The values of b opt;a1 obtained with the three design methods are plotted against g in Fig. 6(a), which shows that the trends obtained with the three methods are similar and that flows of fluids made of complex molecules exhibit lower b opt;a1 . Moreover, the value of b opt;a1 calculated using the empirical method is much higher than the value predicted by the other two methods for all the fluids.
The fluid dynamic performance of the vane operating with toluene at b t1 ¼ 6:0 calculated with the three methods is plotted against b a1 in Fig. 6(b). From the outcomes of the analysis, it can be inferred that the use of loss models based on first principles allows one to predict values of post-expansion ratio and total loss in   relatively good agreement with those predicted by more computationally expensive CFD-based methods.

Variation of the optimal post-expansion ratio (b opt;a1 ) with primary design variables
In order to assess the influence of design parameters on b opt;a1 , the vane fluid dynamic performance was assessed with the CFDbased method without constraints on solidity, flow angle and total-to-static expansion ratio. The working fluid selected for this analysis is toluene. Fig. 7 show the x CFD variation related to the considered vanes as a function of b a1 for selected solidity values and for prescribed constant values of b t1 and 4 a . As expected, the losses in the vanes with higher solidity are higher compared to those with a lower solidity. This is due to the increase in the wetted surface area of the vane with increasing solidity, which increases boundary layer losses. Additionally, it can be observed that b opt;a1 does not change substantially for different values of solidity. In other words, the solidity parameter should not influence the choice of b opt;a1 (or DoD) however, for the considered cases, a lower values of solidity entails lower losses.

Influence of solidity (s)
The Mach number contours of the vanes with varying solidity are illustrated in Fig. 8. The overall flow characteristics, particularly the fish-tail shock pattern and shock wave boundary layer interaction remain identical across the three blades.

Influence of flow angle (4 a Þ
In this case, the flow angle is varied while the solidity and b t1 are kept constant. Fig. 9 illustrates the variation of x CFD with b a1 for three values of 4 a . It can be observed that the losses in the vanes with higher 4 a are higher compared to those of vanes with lower 4 a . The wetted surface area of the bladed region, see Fig. 10, remains the same, and the nozzle geometry is unchanged as the solidity is constrained to a fixed value. However, the length of the straight section connecting the bladed region with the trailing edge increases with the flow angle, i.e., section ss a -ss te in Fig. 2(c). Consequently, this leads to an increase of boundary layer losses for vanes with higher 4 a . Moreover, the boundary layer blockage effect increases with the increase of the flow angle, thus affecting the resulting boundary-layer and mixing losses. This was confirmed by the computation of the values of entropy production between the sections connecting the adjacent leading edge and the trailing edge of the vanes. These values of entropy production are 4.7, 5.23 and 9.14 J,K À1 for 4 a of 71 , 76 and 81 , respectively.
From the results of Fig. 9, it can also be inferred that the optimum post-expansion ratio increases for larger values of flow angles. This is connected to the fact that boundary-layer losses are higher for increasing 4 a , which can become excessively large in case the expansion ratio in the bladed region is kept at comparatively large values. This is due to the expansion ratio in the bladed region being inversely proportional to the post-expansion ratio. Fig. 10 reports the Mach number contours resulting from the simulation of the optimal vanes for three values of 4 a . It can be seen that the wakes in the vanes with higher 4 a , see Fig. 10(c), are closer to each other compared to those with lower 4 a , see Fig. 10(a), leading to higher mixing losses. Moreover, the trailing edge shock reflects on the suction-side surface more upstream in case of 4 a ¼ 81 + if compared to the case with 4 a ¼ 71 + , see   It can thus be concluded that b opt;a1 is highly sensitive to 4 a .  It can be seen that the cubic curve fitting is unsatisfactory for b t1 ¼ 6.0. The choice of the cubic functional form for the fitting derived from the results obtained with the analytical method. The high-fidelity CFD simulations can predict more detailed flow features, therefore the resulting trend of the total losses can be different [11].
Additionally, the losses for higher value of b t1 are larger, since the Mach number in the flow channel increases with the pressure ratio. Moreover, it can be observed that for b t1 ¼ 8:0 a single value of b opt;a1 can be identified, while for b t1 ¼ 6:0 the condition of minimum losses is achieved for a range of b a1 values. This is possibly a consequence of the stronger shock wave occurring at higher b t1 which leads to a higher sensitivity of the overall losses to the post-expansion ratio.
The Mach number contours resulting from the simulation of the optimal vanes for the two values of b t1 are illustrated in Fig. 12. The overall flow characteristics, in particular the fishtail shocks and the shock wave boundary layer impingement, are similar despite the different Mach numbers. These results provide a physical justification regarding the reason why higher losses are computed for the vanes illustrated in Fig. 11, where it can be seen that higher losses corresponds to the case with higher b t1 .

Conclusions
The objective of this work was to investigate the accuracy and thus the suitability of Deych's empirical model for the preliminary design and performance prediction of stator vanes in case the working fluid is made of complex molecules. Deych's method is currently the only one available and adopted in practice for this type of problem. Additionally, the investigation aimed at assessing the influence of primary design variables like, flow angle, expansion ratio and solidity, on the optimum post-expansion ratio of such nozzles.
To this end, the results obtained with the Deych's model were compared with the optimum post-expansion ratio calculated with a physics-based analytical method and that computed with a highfidelity CFD-based method in some paradigmatic cases. Subsequently, the performance characteristics of a set of stator vanes with varying primary design variables was analysed by means of CFD simulations in order to assess its sensitivity to the degree of post-expansion.
The methods were applied to stator vanes operating with fluids ranging from a fluid made of simple molecules to a fluid made of complex molecules. The following conclusions can be drawn: 1. The optimal performance of axial vanes operating at supersonic flow conditions is characterized by a unique post-expansion ratio which depends on the molecular complexity of the working fluid and on the total to static pressure ratio and design flow angle.  2. The performance calculated with the Deych's method is independent of the design flow angle and the method fails to accurately predict the optimum post-expansion ratio for a vane operating at a total to static pressure ratio of 6.0. Therefore, this method should not be used to design supersonic vanes of ORC turbines. A new method to predict the optimum post-expansion ratio of these vanes should be devised. 3. The analytical method described here, see Sec. 2.2, provides values of the optimal post-expansion ratio well in agreement with those given by the high-fidelity CFD-based method (Sec. 2.3) for the examined fluids and operating conditions. This method provided useful insights into the loss mechanisms affecting the expansion process, which, in turn, allowed to justify the unsuitability of the Deych's method for the design of ORC turbine stators. 4. The blade metal angle is found to be the design variable mostly affecting the optimum post-expansion ratio. The effect of the total-to-static expansion ratio of the cascade on the optimum post-expansion ratio requires further investigation.
Future work will be devoted to the development of simplified analytical models for the prediction of the optimum postexpansion ratio starting from data obtained from CFD simulations to be used in the conceptual design phase of supersonic ORC turbine.

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.