Preliminary mechanical design

Key requirements of a turbine test-rig for research purposes are the possibility to test at different speeds and thermodynamic conditions, to obtain accurate measurements of all quantities relevant to performance characterization, to guarantee robust control of start-ups and shut-downs, and to be modular to ease the inspection and assembly of components.

Three main technical aspects were considered in the conceptual design phase:

Figure 4.

Rotor configuration concepts. (a) Straddled impeller, and (b) overhung impeller.

The use of ceramic ball bearings for a turbine of this type provides two main advantages over competing technologies. Firstly, they can withstand substantially higher loads than gas-foil bearings, offering a robust solution for a wide range of operating conditions. Secondly, thanks to the widespread use of REB in a plethora of applications, they are available off-the-shelf and can be readily implemented within a new design. Recent trends in the design of turbocharger systems show that rolling element bearings can be preferable for compact, mobile systems such as automotive turbochargers (Schweitzer and Adleff, 2006), providing, even at high speeds, long term durability with lubrication and mechanical losses comparable to those of noncontact bearings (Nguyen-Schäfer, 2012).

Albeit the use of REB simplifies the assembly design, the need for lubrication poses additional challenges compared to gas-foil bearings. The oil contamination of the working fluid can severely impact the performance and integrity of the entire rig, causing mechanical damage to the rotating components due to erosion. Moreover, leaked oil accumulates and forms sludge, increasing the risk of thermal decomposition of the working fluid whenever in contact with hot spots in the rig. Therefore, the choice of a proper sealing technology is key to ensure safe and reliable operation of rotors supported on oil lubricated ball bearings. To prevent lubricant leakages, a common strategy adopted for large steam power plants is to use dry-gas seals (Forsthoffer, 2011), employing an inert gas like nitrogen as a barrier between the two fluid flows. As an alternative, contact seals capable of guaranteeing almost no oil leak are commonly adopted in low temperature and low speed applications. The advantage of the dry-gas solution is the absence of contact parts between the stationary and the rotating components, but this comes at the cost of a more complex rotor design. The use of dry-gas as a barrier requires also more components and more complex control strategies. Conversely, contact seals lead to a simpler and more compact assembly. The rotational speed and maximum operating temperature of ORC turbines is lower than those of turbochargers, therefore inexpensive contact seals can be used (such as those of the stationary lip seal type). Advancement in the development of low friction materials and temperature resistant coatings enables state-of-art contact seals to operate at peripheral speeds of up to 100ms−1 in combination with temperatures in excess of 250∘C. Therefore, polytetrafluoroethylene (PTFE) based lip seals were deemed appropriate for the ORCHID turbine, and the suitability of this choice was also confirmed by a manufacturer.

The conceptual layout of the rotor configuration obtained as a result of an iterative design process is shown in Figure 5. The rotor includes a shaft (A), a turbocharger cartridge (B), a turbine wheel (C), and a coupling flange (D). A lock-nut on the left side of the bearing inner raceway (E) is used to regulate the mounted-end-play of the rolling elements by adjusting the preload. The cartridge pushes axially against a stainless steel thin-disk (F), whose function is that of moving the oil outwards and to the main drain bore in the center of the assembly, through a cavity realized in the casing and extending 50∘ circumferentially (visible on the right of the main oil drain bore). Stationary parts include two off-the-shelf contact lip seals (G on the impeller side or hot side and H on the coupling side) to contain the oil within the turbine casing, and a noncontact gas seal (J) which was designed in-house.

Figure 5.

ORCHID turbine cross-section, showing the main components and details of the oil delivery system. Red lines indicate the rotating parts. Details K and L are viewed from a cross-section rotated 90∘ with respect to J-J section.

Characterization of the dynamic response of the squeeze-film damper

The main function of the bearing cartridge is to support the powertrain and to transfer the loads to the structure. A three-dimensional representation of the bearing cartridge is shown in Figure 6. The cartridge inner ring, having a bore diameter of 10mm, is fitted to the turbine shaft, while the outer ring has a radial clearance fit of 50μm with the casing. The cartridge outer diameter is 29mm and the overall length is 62mm. Two rows of rolling elements are interposed between the two rings. The outer ring is axially locked in the casing using a snap-ring (item K in detail L of Figure 5), while a rectangular slot and pin system (L) is used as a circumferential lock. The two regions highlighted in red in Figure 6 act as squeeze-film-dampers. Both feature an axial length L of 9mm. The clearance fit ensures that, as a result of lateral motion of the powertrain, the outer ring squeezes the oil-film in the gap, resulting in film pressurization and system damping. The lubricant selected for the application is an ISO VG 32 oil. The oil enters the SFD lands from the supply grooves shown in Figure 5 and is discharged at the other extremity. The bearing cartridge must be operated with a relatively large oil flow rate to cool down the rolling element bearings during their operation. Therefore, the bearing cartridge is considered to be operated in an oil bath and air ingestion is neglected. As a first approximation, ambient pressure is assumed as boundary condition at the two axial ends of the SFD for the investigation of this preliminary design.

Figure 6.

Housing cut-off revealing the rolling element bearing cartridge. The red areas indicate the SFD lands.

The Reynolds equation describes the dynamic pressure generation in the oil film of the SFD. Considering the SFD radial clearance g=50μm, the synchronous excitation ω at the maximum machine operating speed, and the oil density ρ=852.7kgm−3 and viscosity μ=16.3⋅10−3Pas at 50∘C, the squeeze Reynolds number Res=ρωg2/μ<2 throughout the entire range of considered rotational speeds. Previous studies on the development of dynamic forces in SFD (San Andres and Vance, 1987; Delgado and San Andres, 2010) have shown that fluid inertia has a marginal influence on the generation of the pressure field if Re≤10, and it was therefore neglected in this work.

The computer program developed and documented by Gheller et al. (2022) was modified to model the oil flow squeezing within the SFD. The program performs the numerical integration of the Reynolds equation

where θ, z and R are the tangential, axial and radial coordinates, h is the oil-film thickness, P is the oil pressure, and β is the fluid bulk modulus. η and ξ are a so-called complementarity pair, thus variables introduced to linearize the change of fluid properties across phase interfaces. Due to the high operating speed of the turbine and the small radial gap of the SFD, oil cavitation may occur (Zeidan and Vance, 1990). The motion of the cartridge towards the casing causes the oil-film pressure to rise locally due to the squeezing, while on the opposite side of the cartridge the oil-film pressure decreases as the cartridge moves away from the external casing. As a result, the liquid is forced to move circumferentially away from the location of minimum film thickness. Thus, at high pressures the oil flow velocity can be such that its static pressure drops below the saturation pressure and cavitation bubbles form in the film (Nguyen-Schäfer, 2012). The application of the linear complementarity problem (LCP) included in Equation 1 to model cavitation in SFD has been previously demonstrated by Fan and Behdinan (2017).

The powertrain design does not include a centering element for the external ring of the bearing cartridge. Thus, when the machine is not running the external ring of the bearing cartridge rests on the casing. As the rotational speed increases, the rotor excitation (e.g., residual unbalance forces) drives the outer ring far from the casing and generates the squeeze responsible for the oil pressurization. The ORCHID turbine is designed to operate at constant speed, and once the operating speed is reached, the motion of the outer ring reaches the so-called limit cycle. Full characterization of the limit cycle would require a transient simulation of the startup. At the level of preliminary analysis, a parametric study of the rotordynamic performance considering circular orbits and a vertical static eccentricity is conducted. Thus, the relation describing the oil-film thickness distribution over time and space in the tangential direction can be expressed as

where h is a function of the dynamic eccentricity, which in this work has been included using circular orbits of radius e in an orthogonal plane, and of the static eccentricity es with phase θs.

Equation 1 is numerically solved to compute the pressure distribution in the SFD for several shaft positions within a circular orbit. The orbit-based dynamic force coefficients are calculated as suggested by San Andres and Jeung (2016). Figure 7 illustrates the circumferential distribution of the pressure evaluated at the SFD midplane z/LSFD=0.5 and for increasing rotational speed, considering a relative orbit radius of er=e/g=0.25 and a vibration frequency that is synchronous with the rotor rotational speed. As expected, only a fraction of the SFD circumferential extension provides a significant reaction force during operation. This is confirmed by the relatively narrow region where high pressure in the oil-film is developed, whereas the remaining portion of the oil-film undergoes cavitation – i.e., the region where the pressure profiles becomes a flat line – with a pressure of 1,000Pa. The same figure also shows that increasing rotational speeds have a significant impact on the generation of oil-film pressure, highlighted by the increase in magnitude of the pressure peak and by the reduction of the region unaffected by cavitation along the circumferential direction.

Figure 7.

SFD pressure profile at midplane for er=0.25 calculated with equation 1.

The impact of the rotor eccentricity and of the orbit radius on the SFD direct damping and stiffness coefficients is displayed in Figure 8 for the design rotor speed of 100krpm. The two maps displayed on the left of Figure 8 show the variations of the direct stiffness coefficients, while the two on the right show the direct damping coefficients. Several SFD orbits have been simulated with relative static eccentricity ranging from 0 (centered orbit) to 90% of the available radial clearance in the downward vertical direction. A purely vertical and downward shift of the center of rotation is a realistic condition during operation, in which the rotor is off-centered under the action of its own weight. The lower value of the orbit radius is limited to the 5% of the radial clearance. The maximum value of the orbit radius is linearly varied as a function of the eccentricity, up to 90% of the radial clearance for a centered orbit and to 5% for a 90% eccentric orbit.

Figure 8.

SFD stiffness and damping direct coefficients at 100krpm as a function of the relative rotor eccentricity (es/g) and of the relative orbit radius (er).

Considering the left contours of Figure 8, one can observe that the SFD generates increasing direct stiffness coefficients as either the orbit radius and the rotor eccentricity become larger. A squeezed oil-film interested by partial cavitation does not only generate a damping force, but also a centering force (Vance, 1998): the oil-film thus acts as a centering spring. With reference to the right contours, the damping provided by the SFD increases for either small and large orbit radii. This is due to the fact that, despite the portion of oil-film undergoing cavitation increases with larger orbits, the maximum pressure in the film also increases and compensates the detrimental effect of cavitation. As expected, the static eccentricity has a larger impact on kyy and cyy, since a vertical static eccentricity is considered and the outer ring operates with a reduced clearance in that direction. kxx and cxx are instead significantly less affected by the static eccentricity. For the cases of null static eccentricity, centered circular orbits lead to equal direct force coefficients and thus cxx=cyy, and kxx=kyy. Since the static eccentricity considered is purely along the vertical direction, the calculated cross-coupled force coefficients are negligible.

Previous study on oil cavitation in SFD (Zeidan and Vance, 1990) showed that once the so-called bubbly region is reached, oil pressure and thus damping characteristics stabilize even with a further increase of the vibration frequency. Moreover, larger SFD vibrations (thus larger orbits) cause larger pressure differences within the oil-film (San Andres and Jeung, 2016), causing larger portions of the film to cavitate. The dynamic response of the SFD has been analyzed for synchronous excitation with the rotational speed of the rotor. The evolution of the SFD dynamic coefficients with the rotational speed n is shown in Figure 9. Grey bands refer to the variation of coefficients with relative static eccentricities between 0 and 90%, while the colored bands refer to the variation of rotordynamic coefficients with increasing relative orbit radii for the case of a centered rotor. The mean values as a function of rotor speed used in the rotordynamic analysis are shown in the figure by means of a dashed black line.

Figure 9.

SFD dynamic coefficient range for several values of orbit radius and eccentricity. The gray shaded bands are used to account for the variation of coefficients with relative static eccentricities between 0 and 90%, while the colored bands refer to the variation of rotordynamic coefficients with increasing relative orbit radii between 5 and 85% assuming a centered rotor. The dashed back lines indicate the average value.

Design and characterization of a pocket gas seal

The hot side lip seal chosen for this application, i.e., G in Figure 5, can seal up to 5 bar of pressure differential between the bearing cavity and the outside environment. Vice versa, even small pressure gradients would cause the entrainment of gas in the bearing cavity with negative consequences for bearing life. Thus, to prevent pressure build up on the impeller side of the lip seal, a gas seal was introduced between the impeller and the lip seal itself to cause a pressure drop of the pressurized siloxane behind the impeller disk. The selection of the seal type was performed by comparing the rotordynamic coefficients obtained with different types of seals. In particular, for the ORCHID rotor operating conditions and geometry, the calculation of the stiffness and damping coefficients was based on the experimental data for labyrinth seals (LS), fully partitioned pocket damper seals (FPDS) and honeycomb seals (HS) provided by Ertas et al. (2012), normalized following the approach of Childs et al. (1989). The calculation requires the shaft rotational speed, the seal type, length, diameter, the radial clearance and the pressure ratio across the seal. The pressure ratio across the gas seal of ∼1.7 was obtained as the ratio between the turbine stator outlet pressure, and the maximum allowable pressure on the impeller side of the lip seal to avoid siloxane leakage in the bearing cavity. In this application, since the bearing cavity pressure is assumed equal to the ambient pressure, the maximum allowable downstream pressure of the gas seal is also equal to the ambient pressure. A seal length of 9mm was chosen, which was selected as a trade-off between a sufficient number of pocket rows to limit the leakage and the manufacturing constraint deriving from the minimum material thickness separating adjacent pocket rows, while maintaining acceptable overhang of the impeller wheel. The internal diameter of 12.25mm of the gas seal was selected to leave enough radial clearance – corresponding to 125μm in the as-built state of the parts – between the shaft outer surface and the seal inner surface. The value of clearance was informed by a radial stack-up analysis of the components, considering the effect of centrifugal stress, Poisson deformation, thermal expansion, and the contribution of rigid lateral vibrations of the rotor about its barycenter. The maximum shaft diameter of 12mm was chosen to keep the shaft surface speed at off-design conditions below 100ms−1, as prescribed by the presence of the lip seal. The dynamic coefficients computed for three seal types are reported in Table 2. The results of the calculation revealed that the LS provides the lowest effective damping – defined as ceff=(c−k/ω), where ω is the shaft angular speed in rads−1 – with the FPDS achieving similar damping performance as the HS.

To minimize the destabilizing effects associated to traditional labyrinth seals and to avoid the geometrical complexity of FPDS and HS type seals, a pocket gas seal type (PGS) was designed. The PGS is essentially a labyrinth seal with internal swirl breakers, providing additional damping and dynamic stability as in FPDS and HS. The component was designed as a single-body, hollow disk made of stainless steel (AISI 316) with four rows of pockets acting as labyrinths. Adjacent rows of pockets were shifted relatively to each other to avoid preferential paths for the leaking vapor. The pockets extend 50∘ along the circumferential direction and 1 mm along the axial one to guarantee feasibility with additive manufacturing methods.

The pocket seal geometry and its relevant dimensions are illustrated in Figure 10. Additive manufacturing was chosen as production process for this component to allow geometrical freedom and to make the manufacturing cost-effective. Thin walls between adjacent pockets are arguably the most critical component feature from a manufacturing standpoint, therefore their thickness was chosen according to standard design guidelines for 3D metal parts (Ewald and Schlattmann, 2018). Novel printing techniques such as metal binder jetting are effective for manufacturing complex geometries, without requiring additional material for supporting the overhanging regions of the part (Blunk and Seibel, 2023). Nevertheless, material finishing by means of conventional manufacturing techniques will still be required to achieve the small dimensional tolerance required at the inner diameter of the PGS.

Figure 10.

Geometry of the PGS showing the main dimensions in mm.

RANS simulations of the flow within the PGS were carried out to estimate the leakage mass flow and the dynamic coefficients (i.e., stiffness and damping) at the nominal operating conditions of the rotor. A commercial CFD solver (Ansys Inc., 2021) was used for this study, and the computational domain displayed in Figure 11 was discretized using a hybrid approach comprising an unstructured mesh body and a structured layer near the walls. The domain was extended at the inlet and outlet of the PGS-shaft clearance gap as shown in the figure to prevent boundary conditions from influencing the flow solution inside the domain. The fluid domain geometry has been simplified to reduce the computational cost of the simulation, and because the objective of the analysis is to predict the flow motion within the clearance between the PGS and the shaft. Thermal deformation of the seal geometry due to temperature gradients has been neglected, as a result of the minimal temperature drop of just 3∘C across the seal, which justifies the assumption of isothermal operation of the component. The presence of flow swirl at the inlet of the PGS was also disregarded, because a proper characterization of this quantity would require the complete modeling of the secondary flow passage behind the impeller disk. The additional modeling and computational efforts are deemed redundant given the low values of the dynamic coefficients of the seal. Moreover, the uncertainty of the PGS rotordynamic coefficients deriving from this simplification has been included in the form of a sensitivity analysis of rotor stability to the magnitude of cross-coupled stiffness terms.

Figure 11.

Computational domain used for the CFD flow simulations of the PGS. The employed boundary conditions are also indicated. The flow through the geometry of the gap between the shaft outer surface and the PGS inner surface is modeled.

The discretized computational domain comprised 5.1 million cells as a result of a mesh independence study. Local inflation was added at non-slip walls (i.e., shaft and PGS surfaces) to properly resolve the boundary layer, while the dimension of the cells was adjusted in the clearance region to maintain the maximum ymax+<10. Total pressure and temperature were imposed at the inlet and static pressure at the outlet, whose values are reported in Table 3. The SST k-ω model was employed as turbulence closure and look-up tables generated using a multi-parameter equation of state (Huber et al., 2022) were used for computing the fluid properties. Flow simulations for shaft rotational speeds varying from 20krpm to 100krpm were carried out to investigate the impact of the gas seal on system stability at different speeds. A nominal radial gap between the inner seal and the shaft outer surface of 50μm was imposed, which corresponds to the minimal residual clearance estimated for the parts during operation, and also leads to the largest dynamic forces.

The leakage mass flow rate of siloxane MM computed from CFD was compared with the predictions from an empirical method commonly employed in gas turbine engines (Ludwig, 1980), namely the Egli model of labyrinth seal flows (Egli, 1935). Figure 12 shows the comparison of the massflow rate through the PGS predicted by the Egli model and by the CFD as a function of radial clearance c. Despite the Egli model is rigorously valid for labyrinth seal geometries, the mass flow predicted by the two models is in close agreement and varies linearly with the increase in the radial clearance.

Figure 12.

Leakage massflow rate through the PGS as a function of the radial clearance. Comparison between CFD results (×) against predictions with the Egli equation (solid line).

For the estimation of the rotordynamic coefficients, the forces in the horizontal and vertical directions acting on the shaft, Fx and Fy, were calculated by integration of the pressure field on the shaft outer surface. The direct and cross-term stiffness and damping coefficients were calculated according to their definitions

where the denominators are the imposed displacements and displacement rates, whose magnitude was determined by means of a convergence study in which their value was varied until stiffness and damping coefficients reached asymptotic values for small amplitude vibrations. Fx0 and Fy0 are the fluid dynamic forces with null shaft displacement. Figure 13 shows the computed stiffness (see 13a) and damping (see 13b) coefficients at varying rotational speed of the rotor shaft.

Figure 13.

Stiffness (a) and damping (b) coefficients computed with CFD simulations. Dotted lines are linear fits used as input for the rotordynamic analysis. The symbols indicate direct (◯) and cross-coupled (△) coefficients.

At low speed, direct stiffness coefficients are the responsible for most of the PGS stiffness, while the contribution of the cross-coupled stiffness is important only at higher rotational speeds. According to Ertas et al. (2012), seals are characterized by cross-coupled stiffness coefficients opposite in sign which can lead to rotor instability. This is also the case for the PGS considered in this work. The terms on the secondary diagonal of the stiffness matrix are of opposite sign, though their magnitude is similar, i.e., |kxy|≈|kyx|. The magnitude of the generated damping coefficient is also relevant for the evaluation of stabilizing or destabilizing effects on the rotor. It is found, however, that the magnitude of all the damping terms is negligible.

The rotordynamic coefficients computed with the CFD model have been compared to those obtained using the empirical correlation by Ertas et al. (2012) for the honeycomb seal type. Even if the geometry is different, the empirical correlation provides results similar to those obtained with CFD, as reported in Table 4. The prediction of the cross-coupling stiffness by means of the two models differs by about 6.9%. The direct stiffness predicted by the empirical model is about 2.6% higher than the value computed using CFD. Recently, Delgado et al. (2022) conducted a similar CFD investigation to characterize the rotordynamic coefficients of a fully partitioned pocket damper seal (FPDS) and found that the direct stiffness terms computed via CFD are overpredicted if compared to experimental measurements. This result is relevant in that CFD can be used to attain conservative values of the stiffness terms related to the PGS.

Determination of the impeller fluid-structure interaction forces

Fluid-structure interaction (FSI) forces in fluid machines are often responsible for rotordynamic instabilities, potentially damaging the components and in some cases leading to catastrophic failure of the entire system. Unsteady flows in rotor-shroud clearances due to unsteady pressure fields and rotational eccentricities are responsible for the generation of non-axisymmetric pressure fields that cause forces perpendicular to the axis of rotation (Alford, 1965). San Andres (2023) performed an extensive review of the different methods to predict the magnitude of destabilizing forces due to this dynamic forcing using both semi-empirical methods, such as those of Alford (Alford, 1965) and Wachel (Wachel and von Nimitz, 1981), and more advanced methods based on CFD (Moore et al., 2010).

In this study, the destabilizing aerodynamic force coefficient has been predicted according to the Wachel formula (Wachel and von Nimitz, 1981), i.e., the standard method for radial expander-compressor systems (API, 2002). The model allows one to estimate the stiffness coefficient contributing to the systems lateral vibrations

where W˙ is the mechanical power in W, D and b are the turbine wheel inlet diameter and inlet blade height in m, n is the angular speed in rpm, and ρin/ρout is the density ratio across the impeller. In the API 617 (API, 2002) standard, Bc=3, and the value is obtained dividing by ten a fluid molecular weight of 30kgkmol−1, which is representative of light hydrocarbons used in industrial processes.

The destabilizing cross-term was computed for the different operating points of the turbine, using the fluid density ratio and the stage power determined by means of CFD simulations at different operating conditions and plotted in Figure 2. Solid lines in Figure 14 depict the values of kxy obtained from Equation 4 with Bc=3 as function of the stage inlet total pressure and at different rotational speeds. The dashed lines were obtained computing kxy according to the same model but with a modified constant Bc – equal to 16.234 considering that the molecular weight of siloxane MM is 162.34kgkmol−1 –, while the dotted lines were obtained using the correlation developed by Moore & Ransom (Moore et al., 2010) for centrifugal compressors, adapted to radial turbo expanders as suggested by other authors (Lerche et al., 2013; San Andres, 2023). The values of kxy obtained with the modified Wachel model, i.e., for Bc=16.234, and the Moore & Ransom method are similar in magnitude, although the trends as a function of the stage inlet total pressure and rotational speed are qualitatively different. As highlighted by previous research on this topic (San Andres, 2023), a general consensus on the physical phenomena driving aerodynamic instabilities in rotating equipment, and consequently their modeling, is still lacking. Therefore, these estimates can only be used for the selection of the most conservative kxy for preliminary design purposes. Nevertheless, a full characterization of the actual destabilizing effect of the impeller would require at least data obtained with unsteady RANS simulations (Lerche et al., 2013; Pan et al., 2020), and preferably experimental verification on a fully instrumented system.

Figure 14.

Cross-coupled stiffness coefficient kxy computed using the Wachel method of API 617 (solid lines), the modified Wachel formula for siloxane MM (dashed lines) and the numerical correlation of Moore & Ransom (dotted lines) as a function of the stage inlet total pressure for three different rotational speeds.

Consequently, the largest computed value of |kxyaero|=|kyxaero|=6.25×104Nm−1 is used in analysis of the lateral vibrations documented in the following. Notably, these terms of the stiffness matrix are assumed of equal magnitude and opposite sign, and are thus source of dynamic instability for the shaft-train. Subsequently, a parametric analysis as suggested in the API 617 norm, is conducted to assess system stability throughout the operating range of the machine.