Grid generation methodology and CFD simulations in sliding vane compressors and expanders

The limiting factor for the employment of advanced 3D CFD tools in the analysis and design of rotary vane machines is the unavailability of methods for generation of computational grids suitable for fast and reliable numerical analysis. The paper addresses this challenge presenting the development of an analytical grid generation for vane machines that is based on the user defined nodal displacement. In particular, mesh boundaries are defined as parametric curves generated using trigonometrical modelling of the axial cross section of the machine while the distribution of computational nodes is performed using algebraic algorithms with transfinite interpolation, post orthogonalisation and smoothing. Algebraic control functions are introduced for distribution of nodes on the rotor and casing boundaries in order to achieve good grid quality in terms of cell size and expansion. In this way, the moving and deforming fluid domain of the sliding vane machine is discretized and the conservation of intrinsic quantities in ensured by maintaining the cell connectivity and structure. For validation of generated grids, a mid-size air compressor and a small-scale expander for Organic Rankine Cycle applications have been investigated in this paper. Remarks on implementation of the mesh motion algorithm, stability and robustness experienced with the ANSYS CFX solver as well as the obtained flow results are presented.


Introduction
Sliding vane machines are widely employed in industry nowadays. Typical applications where these devices are one of the leading technologies are compressed air systems, automotive thermal management (oil and cooling circuits), oil and gas applications. Nonetheless, unlike other positive displacement technologies that can rely on the design tools at the state of the art to improve and accelerate R&D and product development tasks, sliding vane machine designers cannot entirely benefit from these resources. In fact, the limiting factor for the employment of advanced 3D CFD tools in the analysis and design of rotary vane machines is the unavailability of methods for generation of computational grids suitable for fast and reliable numerical analysis.
In the literature, scientists and engineers have been investigating sliding vane devices both through experimental and numerical approaches. In particular, the modeling approaches that have been developed mostly consider lumped parameter formulations of the conservation equations that undoubtedly benefit of reduced computational cost [1][2][3][4]. On the other hand, they do not allow possibilities of identifying and optimizing key issues in positive displacement machines such as the leakage paths, which in these models are schematized as equivalent orifices [5].
In recent years, the lack of a standardized and parameterized approach to discretize the fluid volume that stator, rotor and blades identify in a sliding vane machine has been started to be addressed by an increasing number of research works. Most of the developed procedures focused on expanders for waste heat to power generation based on an Organic Rankine Cycle architecture (ORC). For instance, Kolasiński and Błasiak performed experimental and numerical studies on a micro ORC sliding vane expander with circular stator and radial inlet and outlet ports [6] while Montenegro et al., investigated a sliding vane expander with an elliptic stator [7]. Working fluids were R123 and R245fa while CFD solvers employed in the afore mentioned studies were ANSYS CFX and OpenFOAM respectively. In both the research works, correction terms had to be introduced to deal with inaccuracies in the rotor grids. In addition to these research works, in commercial software such as Simerics PumpLinx or ANSYS there are templates on vane machines [8,9]. However, they introduce a rather severe approximation when considering a flat blade profile. In fact, together with the stator, the tip of the blade forms a converging-diverging nozzle whose local curvature affect the overall leakage.
Among the available approaches for grid deformation, user defined nodal displacement was found to be the most suitable strategy to create a set of grids representing nodal locations for each time step externally, prior to the numerical flow solution in the calculation domain [10,11]. User defined nodal displacement methodology was successfully applied to screw machine geometries and led to the development of two different strategies for grid generation. An algebraic approach was pursued by the research group at City, University of London [12] while a hybrid differential approach was considered at the University of Ghent [13]. Both the grid generation methods allowed to perform three-dimensional single and multi-phase simulations on screw compressors and expanders in recent years. In particular, the algebraic approach revealed more suitable for extension of grid generation to variable lead and variable profile rotors, while the hybrid differential approach resulted in better grid smoothing and adaptability to variety of rotor profiles such as hook and claw type positive displacement machines [11].
Regardless of the technology, positive displacement machines involve similar issues in the development of a deforming grid generation i.e. from clearance regions to the main core the aspect ratio of the geometry is 1000 th of order of magnitude, the computational cells need to maintain continuity through these highly deforming volumes and the cell structure needs to be maintained in order to avoid solution interpolation from one time step to the other. Hence, in the current study some of the algebraic algorithms developed for screw machinery have been applied to a sliding vane machine geometry and this led to the development of a novel general grid generation methodology. This innovative meshing procedure was tested with reference to two applications that highly differ for dimensions, operating parameters, working fluids and mode of operation. In particular, test cases presented in this paper are an oil injected air compressor and an ORC expander. After recalling the main steps of the grid generation for vane machines, the paper aims at highlighting similarities and differences occurred in the two simulation sets.

Grid Generation Procedure
In a sliding vane geometry, the working chamber is formed by the volume entrained between the housing, the rotor and the end plates. As the rotor carrying the set of vanes turns, each vane divides the full volume into successive chambers that undergo suction, compression/expansion and discharge phases. At the tip of each vane there is a thin gap that needs to be treated as the leakage. Even though the change in volume of this gap is very small compared to the core region, because of the eccentricity between rotor and stator, the relative shape of these gaps change with rotor positions. Altogether, the combined core and leakage gap form a O grid topology. Several algorithms for the structured meshing of O type domain topology are available in practice showed that algebraic algorithms with transfinite interpolation, post orthogonalisation and smoothing are the most efficient ones for applications like twin screw compressors where a number of mesh files are required to be generated and supplied to the CFD solvers in order to treat mesh deformation [14]. Similar algebraic grid generation principle was used in this work for the vane geometry rotor domain mesh.
Flow chart in Figure 1 lists the main stages of the grid generation procedure. A standalone parametric geometry program built as a Matlab application is used to produce the rotor profile along with the vane at the initial rotor position. For a given set of rotor, stator diameters and number of vane with vane thickness, the program provide coordinate of the points on the chamber boundaries. This profile is an input to the meshing algorithm. Successive rotor positions appear to be rigid body rotations for the rotor while for the fluid volume that is deforming, it is a complex boundary motion where rotor rotates, stator is stationary and each of the vane is undergoing a general body motion. For simplification, blade tip clearance gap changes were not accounted in the presented analysis, but the meshing algorithm allows the possibility to further introduce a gap variation function.

Figure 1. Vane geometry mesh generation procedure
The first stage in the meshing procedure is to discretize the boundaries. At this stage, inputs regarding the number of nodes and parameters for various control functions are taken from the user. The boundary discretization produces the distribution into the topology of an 'O' grid, as shown in Figure 2.b. An 'O' structure for the rotor mesh was purposely selected because it avoids any inaccuracies that can be introduced due to a non-matching mesh connection between core and leakage regions. The sudden transition from the leakage gaps to the core was handled by the introduction of stretching functions that gradually flare the radial mesh lines from leakage gap into the core as seen in Figure 2.c and Figure 2.d.
At the position where the core region topology does not remain rectangular, as it happens when the vane slides inside the rotor, a simplification has been introduced to convert the vane side walls into short segments on the rotor surface. It is anticipated that the leakage flow will not be influenced by this simplification as the gap clearance does not change but it significantly eases the O grid mesh generation.
The second stage in mesh generation is distribution of interior nodes. Transfinite interpolation, orthogonalisation and smoothing as described in [14] were used in these procedures to get a good quality hexahedral cell structure as seen in Figure 2.e. Once the 2D mesh is generated for all rotor positions, the mesh is assembled into 3D volume in the required format of the solver (Figure 2.f). In the current implementation there is possibility to export the mesh in solvers such as ANSYS CFX, ANSYS FLUENT, STAR CCM+ and Simerics PumpLinx. The entire meshing procedure was implemented in the software package SCORG™ developed at City, University of London [16].
The rotor domain is eventually connected to suction and discharge ports via non-conformal interfaces, as usually occurs in CFD analysis on positive displacement machines [12]. Ports meshing is usually carried out using standard tools and with reference to a tetrahedral cell structure to better capture the real geometry.
In the presented analysis, ANSYS CFX solver was used to calculate the oil injected flow fields in the air compressor in the first test case and a real gas fluid definition was used in the calculation of a ORC expander in the second test case. ANSYS CFX provides a model called Junction Box Routine that is a user defined library to specify mesh deformation from custom applications such as SCORG. The solver updates the nodes coordinates from set of pre-generated coordinate files after every crank angle step (or its submultiples). Solver time step size finally results from the selected crank angle step and revolution speed of the rotor.

Numerical setups
Two test cases were analyzed using the new grid generation procedure developed in this work as represented in Table 1. rate was 0.203 kg/s, compressor indicated power was 68.2 kW and shaft power recorder was 82.8 kW. Table 1 presents a summary of the geometry and CFD model mesh data for the case. Table 1 also presents a summary of the geometry and CFD model mesh data for the expander case. From the tests performed in refs. [17,18], the operating conditions that were considered as benchmark for this case were 1551 RPM as revolution speed and 12.11 bara as inlet pressure with 90.5 °C as inlet gas temperature. Outlet pressure was 4.57 bara. Aungier Redlich Kwong (ARK) real gas model was used to specify the coefficients of the polynomials to define material properties and equation of state. Coefficients for specific heat and critical state temperature and pressure were supplied to generate a thermodynamic table in the temperature range of 273 and 400 K, while the pressure range was 2.0 and 60.0 bara.  Figure 3 shows the tetrahedral mesh used for the static ports of the oil injected compressor. Suction port, discharge port interface with the deforming rotor domain and the oil injection rail have been highlighted as in the colour scheme. Low pressure inlet and high pressure outlet boundaries are also seen. Similarly, Figure 4 represents the tetrahedral mesh used for the static ports of the real gas ORC expander. High pressure inlet, Low pressure outlet and the axial interface of low pressure port with the deforming rotor domain has been highlighted by the colour scheme. Radial interface of the high pressure port with the rotor is not seen. All these interfaces have been defined in the ANSYS CFX solver as nonconformal conservative flux type of fluid-fluid interfaces. Table 2 presents the details of the CFD solver numerical setup. Most settings remained common between the two cases, except for the working fluid and solver relaxation parameters. Additionally since compressor case was solved as an oil injected flow, an eulerian-eulerian two phase solution was calculated. Air was the primary phase and constant density oil as liquid was set as the secondary continuous phase. Further details of oil injected compressor modelling can be found in Rane et al (2016). In terms of difficulty of solution, the compressor flow poses a challenge. In the compressor, the leakage flow through the tip gaps is in opposite sense to the main positive flow. Because of this and also due to the presence of oil phase, a gradual rise in discharge pressure was required to be specified. At the same time the solver relaxation factors were of lower order. On the other hand even though ARK real gas equation of state was defined with fixed table limits, a relatively easy convergence of the solver was observed. This could be due to the lower time step size and also the leakage flow sense in the same  Figure 5 reports the pressure contours in a cross section of the machine at 75 mm with respect to a full axial length of 540 mm. The sealing of the high pressure port from the suction port is apparent from the pressure distribution. The last 6 th chamber is open to the discharge pressure of 8.7 bara but shows a slightly higher internal pressure of about 10.0 bara due to continued compression of the gas being discharged. Figure 6 reports the distribution of oil volume fraction in the same cross section at the same rotor position. From the leakage paths, oil is getting passed into the 1 st suction chamber on the low pressure side. High oil concentration is visible in the injection rail and a radial jet of oil flow is visible in the 5 th compression chamber. The pressure in this chamber is about 4.0 bara. About 6 -8% oil volume fraction is observed in the discharge port. Figure 7.a presents the distribution of gas temperature in this cross section while Figure 7.b is a 3D distribution of gas temperature with an oil iso-surface with 10% volume fraction superimposed in greyscale colors. Despite the sealing action of the lubricant, the clearance at the vane tip induces a backflow driven by adverse pressure gradient in opposite direction to the counter clockwise revolution speed sense in Figure 7.b. In turn, the overall flow field in the compressor vane is affected by a recirculation region whose magnitude is gradually lowered as the leakage flow interacts with the main one (which is aligned with the rotation sense). This phenomenon contributes to enhance the heat transfer between the compressing air and the oil that, after the radial injection shown in Figure 7.a, spreads along the leading side of the vane and mixes with air. The oil accumulation on the walls of the rotor, i.e. on the metallic surfaces which define the vane in reality, can be seen in the bottom left of Figure 7.b. These oil layers allow to lower friction losses between surfaces, namely stator with blade tip and blade side walls with rotor slots.  Figure 8.a reports the pressure distribution of gas in case of the ORC expander simulation. Pressure from the high pressure port is at 12.11 bara and 90.5 °C. With expansion, a smooth decrease in gas pressure is visible across the chambers. Finally, pressure is at lower level of 3 bara from the 4 th chamber onwards, when low pressure port is exposed. Figure 8.b represent the pressure distribution on the 3D rotor surface along with the vanes. A uniformity of pressure distribution is seen is individual chambers. Across the vane tip leakages, pressure gradients are observable. Similarly, Figure 9.a reports the gas temperature in the cross section and Figure 9.b presents gas temperature on the 3D rotor surface. In the cross section gradual reduction in gas temperature is visible. On the 3D rotor surface in chambers 1 and 2, a small non-uniformity of gas temperature distribution is seen due to the length of the intake port not covering the full rotor length and presence of axial low pressure ports on the two axial ends of the rotor.   Table 3 presents the computational time required for generation of the grids using the developed algorithm and the time required by the CFD solver for calculation of a one boundary condition.

Conclusions
The research work presented in this paper is the development of a robust and fast, deforming grid generation procedure for CFD analysis of sliding vane machines. Available literature suggested that only limited approaches to solving a full 3D numerical simulation in vane machines is currently available and the major limitation was found to be in the generation of deforming grids for the rotor domain. The developed grid and numerical methodology was eventually tested using the ANSYS CFX solver. Two vital test cases have been reported in the paper. First case is an industrial sliding vane oil injected air compressor and the second case is an ORC application, R236fa operated expander.
Multiphase simulations on the oil injected air compressor were successful in revealing the nonuniformity of gas temperature and oil distribution during the operation of the machine. Such a full scale model allowed to track a recirculation region induced in the vane flow field by the leakage flow through the clearance at the tip. Furthermore, the convective heat transfer with air showed the cooling capabilities of the lubricant even when supplied into the vanes as liquid jets. On the other hand, the oil temperature rise decreased its dynamic viscosity and, in turn, its hydrodynamic potential to prevent dry contact between blades and stator. Real gas ARK model was successfully applied in the ORC expander analysis. The simulations provided information on the expander filling losses, internal expansion phenomenon and performance parameters could be evaluated at various operating conditions. Although due to the scope of this paper, flow results only from one of the selected operating conditions have been presented here.
These new procedures for grid generation of sliding vane machines will allow further design improvements and performance prediction of the machines with design modifications can be performed better. The effects of a variable tip clearance due to oil temperature and blade dynamics will be taken into account in future upgrades of the numerical methodology.