Effect of Lattice Structures in the Stress–Strain State for an Impeller Turbine

Abstract

The stress level of a rotating component is of vital importance in order to ensure its safe operation. The primary source of stress for this type of component is the induced centrifugal stress, which depends on the material, rotational speed, and the distribution of the mass. The reduction of stress has been a topic of study for some time; however, the advent of additive technologies has prompted a new wave of research into the design and manufacture of centrifugal impellers for gas turbine engines, incorporating internal lattice structures (LSs). These structures offer benefits in terms of material savings and load reduction by decreasing the centrifugal force. The present work analyzes the stress–strain state of a turbine centrifugal impeller for six different designs, distinguished by the presence or absence of LSs of various geometries, achievable only through additive technologies. The analysis was conducted on a turbine impeller, which serves as an example of a promising small-scale gas turbine engine (SSGTE). The effectiveness of LSs was assessed through their unloading effect; furthermore, an approach to identify their optimal location within the impeller was demonstrated.

1. Introduction

Centrifugal impeller turbines represent one of the main components of the small-scale gas turbine engines (SSGTEs). The impellers are responsible for decompressing the airflow at high temperatures, thereby contributing, in this way, to the thermodynamic cycle of the engine [1]. The design of an impeller is a highly complex and critical task that requires the balancing of numerous elements to ensure optimal functionality, safety, and efficiency [2]. The nature of the design intent will determine which factors are fixed and which are treated as variables. These factors are inextricably linked, with one influencing the others in a manner that can be either more or less demanding of the capacities of the other factors. The most critical considerations when designing an impeller are as follows: aerodynamic efficiency, thermal management, material selection, size, mass, rotational speed, vibrations, centrifugal forces, stress analysis (SA), fatigue lifecycle, tolerances manufacturability, assembly constraints, and environmental considerations [3]. Each of the aforementioned considerations has its own set of critical factors, subdivisions, and peculiarities. Each design decision has an impact on the efficiency, durability, and performance of the resulting device.

The principal source of stress within the rotating components of an SSGTE, including the impellers, are the stresses induced by the rotational speed. These stresses represent the primary concern for structural integrity when they are compared with the stresses produced by other factors, such as pressure and temperature, which are relatively low in comparison [4]. It is always preferable to maintain the aforementioned stresses at a low level, in order to avoid any potential catastrophic scenarios. Consequently, engineering efforts are focused on reducing the primary stress source, which is directly related to the mass and rotational speed; the mass depends on the size and material, while the rotational speed depends on operational requirements, such as the aerodynamic performance and temperature. Considering a constraint to fix the size and rotational speed, as well as their related parameters, such as the dimensions, overall shape, aerodynamic performance, and temperature, the mass is the only parameter that can be treated as a variable, to reduce the stresses [5].

It is of the utmost importance to exercise caution when reducing the mass of the impeller, as this may potentially compromise its structural integrity and resistance to operational stresses. It is therefore essential to ensure that the reduction in mass does not result in the weakening of critical areas subjected to high levels of stress. In order to reduce the mass while maintaining or even improving its structural integrity, a number of strategies may be employed [6,7,8]. These include the use of hollow structures, the optimization of blade thickness, the introduction of lightweight materials, the implementation of ribbed designs, the reduction in the radial length, the employment of splitter blades, the elimination of non-essential material, the streamlining of blade profiles and the replacement solid features with lightweight alternatives.

Considering the strategies for mass reduction along with the prior constraints imposed to maintain the operational requirements, it is only feasible to consider the following strategies, the employment of hollow structures and the elimination of non-essential material. These strategies are intrinsically interlinked. It is crucial to understand and manage the excessive stresses, in order to prevent any impeller failure [9]. By considering an axisymmetric stress distribution within the impeller and simplifying the scenario to 2D, incorporating solely radial and hoop stresses, thereby precluding shear stress, the equivalent Von Mises stress at any given axial distance can be derived as follows:

$${\sigma }_{VM}=\sqrt{{\displaystyle \frac{1}{2}}\left[{\sigma }_H^2+{\sigma }_R^2-{\sigma }_H{\sigma }_R\right]},$$

(1)

being, ${\sigma }_H$ the hoop stress and ${\sigma }_R$ the radial stress, which in turn are governed by the following equations [10]:

$${\sigma }_H={\displaystyle \frac{\rho {\omega }^2}{8}}\left[\left(3+\upsilon \right)\left(R_1^2{+R}_2^2+{\displaystyle \frac{R_1^2{R}_2^2}{r^2}}\right)-{\left(1+3\upsilon \right)r}^2\right],$$

(2)

$${\sigma }_R=\left(3+\upsilon \right){\displaystyle \frac{\rho {\omega }^2}{8}}\left[R_1^2{+R}_2^2-{\displaystyle \frac{R_1^2{R}_2^2}{r^2}}-r^2\right],$$

(3)

where

υ is the Poisson ratio;

$\rho$ is the density of the material;

ω is the angular velocity;

R₁ is the inner radius;

R₂ is the outer radius;

r is the radial distance from the axis of rotation.

These equations demonstrate the rationale behind the observation that stress levels increase significantly with higher rotational speeds and larger radial distances from the center of rotation, since they have a quadratic exponent, leaving little room for the mass to contribute to inertial stress.

The previous strategies for mass reduction, when adopted in conjunction with the aforementioned equation, allow for the creation of lightweight impellers that meet the design intent and resist extreme conditions to which they are subjected. This is achieved by ensuring that areas of high stress are not compromised structurally by the mass reduction efforts. The implementation of this in a real impeller design can be challenging and time-consuming, as it is an iterative process. In this regard, advanced simulation tools, such as the Finite Element Analysis (FEA) [11], play a crucial role in the analysis of impellers that can withstand the extreme conditions within an SSGTE, thereby ensuring the design intent in terms of high efficiency and reliability over the engine’s operational lifespan. This is a topic of significant interest for researchers at the present time and has considerable potential for further development.

Several studies have been conducted with the objective of analyzing the stress–strain state of rotative components. For example, Ref. [12] presents an analytical method for the rotor SA, offering insights into the strength of a rotor. Similarly, [13] proposes an analytical formulation to accurately calculate the stress and displacement of a solid cylinder rotor at a high speeds and temperatures. Additionally, research has been conducted to assess the impact of the mass reduction on the stress experienced by rotating components. For instance, Ref. [14] investigated the potential of topology optimization for high-speed centrifugal compressor impellers, obtaining a substantial decrease in both the mass (27%) and maximum stress level (30–40%) compared to a state-of-the-art solid centrifugal compressor wheel.

In work [15], the authors conducted a preliminary exploration of topology optimization for a centrifugal impeller, considering the use of additive manufacturing (AM); their findings demonstrated that it is possible to reduce the mass while maintaining the stress levels at the same level and decreasing the deformation. AM offers notable advantages, particularly its capability to produce components with complex geometries, albeit with certain limitations. However, parts produced via AM often exhibit suboptimal surface quality. As highlighted in the literature [16,17,18,19], the surface quality depends on a variety of parameters, encompassing geometric factors, such as the surface roughness, dimensional deviations, and waviness, as well as physic-mechanical aspects, including the surface hardening depth, residual stress distribution, chemisorption, secondary structure formation, and diffusion from lubricants.

Components created via AM often require post-processing to achieve the desired geometric accuracy and surface finish [20]. However, this step is challenging due to restricted tool access to critical areas. Moreover, the surface layers of such parts are prone to tensile residual stresses, and the material frequently exhibits notable anisotropy in its mechanical properties, collectively leading to a marked decline in the overall performance. Given the challenges outlined, the development and adoption of advanced design methodologies for accurately determining the stress–strain states of components during operation is essential. This need arises not only from increasing product complexity and stricter physical and mechanical property requirements but also from the iterative nature of manufacturing, where production issues necessitate design modifications [21,22]. These methodologies must extend their analysis capabilities to parts produced via additive technologies, encompassing not only material properties and surface quality parameters but also the stress–strain states of design elements unique to additive manufacturing.

A notable example of design elements exclusive to additive technologies is the incorporation of porous or lattice structures (LSs) within a part’s body. Commonly referred to as cellular or honeycomb structures in the literature [23], these features are primarily used as support materials to improve the quality and stability of the printing process. Over time, however, their applications have expanded significantly, addressing a growing array of engineering challenges [24]. The study presented in [25] provides a timeline and illustrative examples of LSs, highlighting their increasing complexity and diversity over time. Consequently, numerous researchers in additive manufacturing, such as those cited in [26], have developed comprehensive classification systems for LSs. However, due to the rapid evolution of additive technologies and the relatively recent introduction of LSs (dating back to the early 2000s, as noted in [27]), new works necessitate the addition of categories to the existing classification frameworks [25,26,28]. Figure 1 showcases the visual characteristics of LSs fabricated using 3D printing with metal alloys.

LSs have a wide range of applications, encompassing diverse functional and aesthetic purposes. They are often used as printing support material, as fillers within a part’s body to reduce its overall mass—and by extension, the mass of the entire assembly structure—or to enhance the bearing capacity of the part by optimizing the load distribution based on the mass distribution. Additionally, LSs are employed for decorative purposes and in the design of medical implants and endoprostheses.

In study [29], a multi-scale concurrent topology optimization of the material and structural design under mechanical and thermal loads is considered. Porous Anisotropic Material with Penalization is employed to distribute material on two length scales in an optimal way. The findings indicate the efficacy of the proposed method. The numerical evidence also suggests that, compared with solid material, porous material with a well-designed microstructure may be a superior choice when thermo-elastic effects are considered.

As previously discussed, the advantages of AM have led to the development of turbine and compressor impellers with LSs. These structures are designed to reduce the loads resulting from the influence of centrifugal forces. The field of research related to the manufacture of rotating components, as exemplified by [30], which focused on defining the manufacturing process of a closed impeller using AM, has resulted in significant advances in the knowledge base, thereby increasing the technological readiness of this kind of process. Additionally, [31] investigated the feasibility of AM for realizing the lattice-based structure of turbomachinery components. A series of studies [32,33,34,35] has been dedicated to the analysis of LSs in an impeller at different stages like the printing simulation, including residual stresses and deformation analysis, static analysis, and lightweight design. These studies have provided an overview of the means by which LSs may be addressed.

This study focuses on the evaluation of the stress–strain state of a turbine centrifugal impeller that belongs to an SSGTE, which is currently being evaluated. The preliminary SA of the impeller indicates that this component requires stress mitigation due to the presence of excessive stress levels at the hub, which is an area of great concern. This study conducts a comparative analysis of potential stress-mitigation alternatives, with the objective of identifying solutions that can be implemented without altering the fixed design and operational parameters. Two strategies are evaluated in this study: incorporating LSs via AM to reduce mass and relieve stress, and reinforcing the hub while optimizing the blade thickness. It is important to note that this analysis is constrained to the structural effects of LSs and does not account for changes in the material properties or AM techniques, which will be addressed in future studies if the LS proves effective.

2. Materials and Methods

One of its main components, of the promising SSGTE that is currently being evaluated for its viability for operational deployment, is the turbine centrifugal impeller, which has 13 blades and the following main dimensions: a height of 141 mm, an external diameter of 204 mm, and an internal diameter of 25 mm. The impeller and its dimensions can be appreciated in Figure 2, which also depicts the internal gears that connect the impeller with the shaft. It is important to mention that this impeller is produced through a casting process, resulting in a continuous internal structure.

The material selected for the impeller is the heat-resistant nickel-based superalloy ZhS6U (K465) (Samara National Research University, Samara, Russia), which is extensively utilized in the fabrication of components exposed to high temperatures within gas turbine engines (GTEs). The chemical composition is presented in

Table 1. Chemical composition of ZhS6U (K465) superalloy.

Cr	C	Al	Ti	W	Nb
8.0–9.5	0.13–0.2	5.1–6.0	2.0–2.9	9.5–11.0	0.8–1.2
Mo	Co	Fe	Ni	Other (Si, S, P, Ce, Zr, B, Pb, Bi, Y)
1.2–2.4	9.0–10.5	≤1	Base	≤0.93

[12]. The material properties employed in the analysis were extracted from [36], with the most important properties being the density of 8420 kg/m³, the Young Modulus, which has values between 1.96 × 10⁵ and 1.28 × 10⁵ MPa for a temperature range of 20 and 1000 °C, and the tensile yield strength, which is 914 MPa.

A preliminary analysis of the stress–strain state for the impeller revealed that the hub is the area of greatest concern, exhibiting considerable stress. In the course of this paper, the impeller in question will be referred to as the “original design”. According to the design intent, the majority of the parameters, like the overall shape and size, were already established as fixed, as were the operational requirements, like the temperature and rotational speed, with no possibility to modify them (these operational requirements will be described in detail in the sub-section of Boundary Conditions). Consequently, efforts were directed at reducing hub stresses without altering these operational requirements.

The objective of this study was to analyze the stress–strain state of the original design and compare it to alternative designs to mitigate hub stress. As part of this initiative, two main strategies were proposed: the first strategy was to incorporate LSs, to reduce mass and achieve an unloading effect; it is important to note that this would need to be performed through AM. The second proposal was to reinforce the hub and to reduce the thickness of specific regions of the impeller. It is crucial to highlight that the present study is focused on the investigation of strategies to mitigate hub stresses in the impeller, through the two proposals previously mentioned. Therefore, the analysis of the designs incorporating LSs is limited to the impact of unloading from cavities and does not examine the changes in the material properties resulting from AM or the distinct types of AM techniques that could be employed to produce the impeller. Should the incorporation of LSs prove to be a promising solution, these options will be explored in a future analysis.

In recent years, various software tools have been developed to support the creation of virtual geometries for AM, often incorporating features to simplify the modeling of LSs. However, this study did not utilize such tools due to limitations associated with Finite Element Models (FEMs), which require solid geometries, while these tools typically produce faceted geometries. Progress is being made to overcome this limitation through more advanced and versatile CAD programs. Nevertheless, for this study, it was chosen to model LSs using a solid virtual geometry of the impeller, employing an established methodology that involves generating a designated virtual volume for the LS and subtracting arrays of repetitive elements.

The subtraction process of layer-by-layer spherical elements from the impeller unloading zone is presented in the Figure 3a, along with an example of a cube resulting from this process. Figure 3b,c also showcase the results of layer-by-layer subtraction using spherical and cubic elements, respectively. For a comparison, Figure 3d depicts an analogous volume from the unloading zone of a GTE blade, designed by the authors in [37].

Consequently, a total of six distinct designs were devised (including the original design), distinguished by the presence or absence of LSs of varying geometries. As a result, the remaining five designs represent efforts to mitigate the stress in the problematic areas. These designs were created using the Siemens NX 12.0 software and are presented in Figure 4 and Figure 5. The second and third designs are analogous to the original design but incorporate the concept of the first proposal. Consequently, they exhibit LS of distinct shapes beneath the surface layer.

The fourth design is analogous to the original design but incorporates the concept of the second proposal. The design incorporates a reinforced hub and the thinning of specific impeller regions. In the course of this paper, the design in question will be referred to as the upgraded design. The fifth and sixth designs reflect the integration of the two main proposals. In this manner, the designs are analogous to the upgraded design and also exhibit LSs beneath the surface layer.

The spherical elements, which were employed in the second and fifth designs and constitute the LSs under study, have a diameter of 6 mm, and the partition thickness between them is 2 mm. The structure in question exhibits a continuity coefficient (defined as the ratio of the actual volume of the LS to the total volume allocated for filling) of 0.779. The cavities also assume the form of bricks, exhibiting a cube-like structure and sides measuring 6 mm. The partition thickness is also 2 mm. The continuity coefficient of this particular structure is 0.578. As illustrated in Figure 4 and Figure 5, the designs also exhibit boundary cavities of an arbitrary shape, which is a consequence of the lack of available space for their accommodation.

Table 2. Mass values of the six designs and the corresponding percentage mass variation.

Design/Parameter	1st Design	2nd Design	3rd Design	4th Design	5th Design	6th Desing
	Original	Original w Spherical LS	Original w Brick LS	Upgraded	Upgraded w Spherical LS	Upgraded w Brick LS
Mass [kg]	5.996	5.590	5.206	5.238	4.865	4.629
Mass Variation [%]	N/A	−6.8%	−13.2%	−12.6%	−18.9%	−22.8%

presents the mass of the six designs and the percentage reduction relative to the mass value of the original design. This reduction was derived from the structural arrangements and the incorporation of LSs in both the original and upgraded designs.

2.1. Boundary Conditions

The force and temperature loads utilized in the FEA, which elucidate the impact of the working fluid on the elements of the impeller turbine, were provided by the authors of work [38]. By employing the developed finite-volume models that incorporate the methodologies of computational gas dynamics, the researchers were able to generate tables containing the values of pressure and temperature at specific points on the surface of the virtual body of the impeller turbine [39]. As the parameters of the working fluid vary in magnitude with different values of the SSGTE rotor speed, the aforementioned research team prepared tables with the values of pressure and temperature at different values of the rotor speed. Accordingly, when evaluating the stress–strain state of the impeller turbine, the variation in pressure and temperature of the working fluid in response to alterations in the rotation speed was considered. It should be noted that the incorporation of temperature data required the development of supplementary FEM, which enabled the generation of temperature field distributions for virtual turbine runner configurations. The aforementioned models permitted the generation of tables with temperature values at points in space limited by the virtual body of the impeller, based on the temperature at the contact zone between the working body and the impeller. For further details, please refer to the studies referenced in [40,41].

In accordance with the technical specifications, the engine operation mode was initially analyzed at a rotation speed of 50,000 rpm. The temperature and pressure of the gas flow at the inlet of the impeller turbine at this rotation speed are 1003 K and 276.9 kPa, respectively. At the outlet of the impeller turbine, the temperature decreases to 794.5 K, while the pressure decreases to 109.8 kPa. The pressure distribution in the impeller original design can be observed in Figure 6, while the temperature distribution can be observed in Figure 7.

Additionally, the initial data for the FEM included the conditions of contact between the impeller and the rotor elements, as well as their kinematics. The data set comprised the force loads resulting from the impact of the working fluid flow on the rotor parts and installation conditions. These details are referenced in sources [42,43,44]. These boundary conditions employed are illustrated in Figure 8, which depict the rotational speed and the locations where the displacements were applied to the original design.

It is important to note that all the boundary conditions mentioned here and illustrated in Figure 6, Figure 7 and Figure 8, were consistently applied to all remaining designs.

2.2. Model Construction

In order to evaluate the stress–strain state of the impeller designs, six FEMs were developed in ANSYS Workbench 2022 R1. Prior to this, three categories of elements were identified within each design variant of the virtual body: blades, rounding zones, and the base of the impeller turbine. The finite element mesh for the blades and rounding zones was created using the “Sweep” method, which is optimal for analyzing products with smooth shape changes and a complex loading pattern. In the context of the Sweep method, the primary finite elements employed are designated as “Wedge 15” and “Solid 226”, the latter of which belongs to the “Hex 20” category. The Sweep method is frequently employed in the development of finite element meshes, as it enables a reduction in the computational time while maintaining the accuracy of the results. Additionally, the Solid 226 elements were employed for the mesh of the impeller hub. The principal advantages of the Wedge 15 and Solid 226 elements include the capacity to employ them in problems that take thermal processes into account. Moreover, these elements contain intermediate nodes situated along its edges and not only at the corners of the element. The presence of intermediate nodes enables a more precise characterization of the surface curvature and deformations of the object, which ultimately results in more accurate outcomes in the FEA. Therefore, the primary objective of employing these elements with intermediate nodes is to enhance approximation in regions exhibiting evolving geometry or intricate temperature or stress gradients. This strategy of dividing it into a finite element mesh has been demonstrated to yield reliable results in a substantial number of studies, as evidenced by references [45,46].

An example of the resulting mesh for the original and second designs, using the methodology described previously, is shown in the Figure 9.

3. Results and Discussion

From Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, it is possible to gain insight into the outcome of the calculations conducted, which also illustrate the distribution of the safety factor across the virtual geometries of the designs.

The primary outcomes of the calculations indicate that filling the impeller turbine body with the LS results in a reduction in its load-support capacity, for the structures under study. This implies that the unloading effect resulting from the mass reduction at the periphery of the impeller has no greater impact than the effect of increasing stresses at the impeller hub due to a decrease in its material amount.

It is noteworthy that the unloading effect is less pronounced in the second and third designs than in the upgraded designs with an identical internal LS that are the fifth and sixth designs, which exhibit a comparatively reduced unloading effect. This discrepancy can be attributed to the design upgrade of the impeller, specifically the transition from the design depicted in Figure 4a to the design depicted in Figure 5a; during this process, the material amount in the periphery was reduced, which subsequently did not allow for the placement of the LS within the same volume as the original design.

Resulting from the applied structural arrangements,

Table 3. Von Mises stresses maximum values, acting in the impeller hub and the corresponding percentage of stress variation.

Design/Parameter	1st Design	2nd Design	3rd Design	4th Design	5th Design	6th Desing
	Original	Original w Spherical LS	Original w Brick LS	Upgraded	Upgraded w Spherical LS	Upgraded w Brick LS
Von Mises stresses ${\sigma }_{VM},\hspace{0.33em}\mathrm{M}\mathrm{P}\mathrm{a}$	938	1000	943	821	860	844
% Stress Variation	N/A	+6.6%	+0.5%	−12.5%	−8.3%	−10.0%

presents the maximum Von Mises stresses that arise in the impeller hub, along with the percentage of stress variation relative to the stress value of the original design, while

Table 4. Safety factor for the six designs.

Design/Parameter	1st Design	2nd Design	3rd Design	4th Design	5th Design	6th Desing
	Original	Original w Spherical LS	Original w Brick LS	Upgraded	Upgraded w Spherical LS	Upgraded w Brick LS
Safety factor, k	0.97	0.91	0.97	1.12	1.07	1.09

presents the values of the safety factor and its percentage of the safety factor relative to the original design.

A correlation diagram was constructed based on the observed reduction in mass as presented in Table 1 and the variation in stress as presented in Table 3. This is illustrated in Figure 16.

In light of the findings of the conducted research, it can be posited that the greatest effect on the reduction of stress was achieved with the upgraded design, whereby the hub zone was strengthened and the peripheral zones were thinned without the introduction of LSs, as illustrated in Figure 5a.

In order to ascertain the impact of stress reduction resulting from a reduction or increase in the mass of the impeller turbine body, a numerical experiment was conducted. In order to achieve this objective, FEMs were developed based on virtual bodies of the original design, but incorporating different cavity volumes. Figure 17 illustrates the results of calculations based on the developed FEM; the volume of the cavity was varied by modifying the “h” size, which represents the distance from the cylindrical surface of the impeller hub to the surface at the cavity bottom.

A curve was plotted based on the experimental results, which demonstrates the optimal cavity position for unloading the impeller. This position achieves the greatest effect by balancing the reduction in impeller stress through a reduction in mass with the increase in stress through a reduction in material quantity in the impeller hub zone. Figure 18 illustrates the changes in Von Mises stress values in the hub zone in relation to the volume occupied by the cavity, expressed as a percentage. Additionally, it depicts the wall thickness between the hub’s cylindrical surface and the cavity’s bottom, designated as “h” (see Figure 17).

4. Conclusions

This work investigates the utilization of LSs in an impeller turbine subjected to centrifugal forces through the use of numerical structural analysis techniques. Based on the results, the following conclusions are drawn.

•

The better balance between mass reduction and stress reduction was achieved with the upgraded design, which resulted in a 12.5% reduction in stress and a 12.6% reduction in mass. This was performed by reinforcing the hub zone and reducing the thickness of the peripheral zones, without the incorporation of LSs.

•

The employment of LSs represents an effective strategy for the reduction of the mass of the impellers. Nevertheless, a reduction in the impeller mass does not result in a corresponding decrease in the induced stresses due to centrifugal forces.

•

The efficacy of LSs is contingent upon the geometry of the components in question.

•

The unloading effect is more pronounced when LSs are positioned in the peripheral regions of the impellers.

•

In the event that the impeller in question exhibits a relatively small periphery volume and the LSs are constrained to the hub zone, the placement of the LS will inevitably result in a pronounced surge in stresses. This is a direct consequence of the diminished strength of the hub zone, which is a result of the aforementioned constraints.

Subsequent research will concentrate on the characterization of the 3D-printed superalloy ZhS6U (K465), which is currently undergoing and actually inspired this manuscript, in order to explore its potential to redefine mechanical and thermal properties, as well as its adaptability for advanced applications, as well as their limitations, like anisotropy and surface finish.

In addition to the aforementioned point, the multi-scale concurrent topology optimization will be contemplated employing the same rationale as in reference [29]. The material will be distributed using a Porous Anisotropic Material with Penalization, as it has been demonstrated to yield satisfactory outcomes. The authors concluded that the utilization of a porous material with a well-designed microstructure is preferable to that of a solid material.