An Optimization of a Turbocharger Blade Based on Fluid–Structure Interaction

Abstract

The structural fracture of the compressor blade is the main cause of fatigue failure. The novelty of this paper is the creative application of bent swept-back modeling to the blade of the turbocharger impeller. This paper is based on a compressor impeller satisfying the k-ε turbulence model. A simulation model was established in ANSYS software, the fluid–structure interaction was calculated in the three models before and after improvement, and the results were compared and analyzed. The optimized blade could improve the blade structure, reduce stress and deformation, and improve the pressurization ratio. In this paper, the optimization scheme of different parameters was discussed in line with the optimal solution. Based on the combination of fuzzy and grey correlation theory, it was concluded that the correlation between pressure and total deformation was higher than that of equivalent stress, and these two values reached 0.8596 and 0.8001, respectively. The results showed that the pressure and total deformation were significantly related to the flow rate. It provides a feasible scheme for further improvement of the supercharger compressor.

1. Introduction

Turbocharging technology is vigorously promoted and applied today, which can greatly improve the power [1], economy, and cleanliness of engines, and it has become one of the key technologies for energy saving and emission reduction of internal combustion engines [2]. Yet, the high speed, high temperature [3], and high vibration of the supercharger not only make it become an important source of engine and vehicle noises but also seriously threaten its reliability [4,5]. As the most important part of the turbocharger [6], the turbocharger compressor blade works at a high temperature and high pressure, bearing centrifugal force and gas power, as well as corrosion, oxidation, and other effects. The blades generate exciting force due to various reasons in practical work. Under the long-term action of excitation force, the noise will be generated at the compressor inlet [7], increasing the possibility of blade damage, and thus affecting the safety of the turbochargers [8].

In the process of impeller high-speed rotation, the flow field is rather considerable to the characteristics of impeller structure, where nearly half of the blade failures are related to it. With the increasing dependence of diversified machinery on turbocharging technology, the optimization of blade structure and internal flow field becomes particularly important.

As an important part of the turbocharger, the compressor impeller has experienced several different stages of development [9]. The original turbochargers consisted of swept-back vane-enclosed impellers and stream-flow single- or two-stage compressors [10]. Since the Second World War, as the research of axial flow compressors has made great progress [11], a few turbochargers have used multistage axial flow compressors for experimental purposes [12]. However, using them as turbocharger components was not attractive from an economic point of view and never reached the point of production [13]. For turbocharging with a supercharging degree above 2:1, due to the limit of strength, the closed impeller of the swept-back blade once disappeared and was replaced by the impeller of a semi-open runoff blade with a shaft guide wheel [14]. After entering the 21st century, due to the further improvement of the degree of pressurization and the demand for a wide efficient flow range, a large number of swept blades with a sweep angle of 20–50° are adopted, which greatly improves the performance of the compressor impeller and turbocharger stage. A performance evaluation test was carried out for the sweep angle γ = 30°, the influence of the leading-edge sweep angle on the performance of the delta blade turbine, and the hydrodynamic characteristics of the flow around the turbine are analyzed. The literature data show that the leading-edge sweep angle influences the turbine efficiency and the hydrodynamic characteristics of the flow around the turbine [15]. The continuous progress of science and technology provides objective conditions for turbo machinery to develop in the direction of high speed, high load, and automation, and also puts forward higher and higher requirements for turbomachinery in terms of speed, capacity, efficiency, safety, and reliability [16,17]. However, turbo machinery often has a variety of faults, such as impeller and blade row being able to resonate with any excitation generated by turbo machinery, affecting its normal operation. Fatigue is often the root cause of rotating parts failure, and sometimes there will be even serious machine damage and fatal accidents, resulting in huge economic losses [18].

In the design method of the turbocharger, the aerodynamics calculation, flow field analysis, and structure design of compressor and turbine are implemented by computer for many years. Subsequently, the aerodynamic calculation, three-way flow field analysis, impeller and blade profile design, and strength analysis of the centrifugal compressor are carried out by computer. At present, major turbocharger manufacturers have advanced CAD/CAM and cat systems, and conduct three-dimensional viscous flow modeling and verification analysis [19].

The root of the blade is usually the point of maximum stress. To reduce the maximum stress here, the structural parameters of the turbine blade can be adjusted [20]. At the same time, the adjustment of the blade aerodynamic model can improve the turbocharger ratio. Naik et al. [21] analyzed the vibration stress of different blade shapes and deduced the relationship between the total stress and the thickness of the blade. The research of Sadanandam et al. [22] showed that to improve the blade bearing higher temperature and pressure in the thermal cycle, the operating speed could be changed appropriately or the blade material could be changed. To optimize the flow field in the compressor, Wang et al. [23] made a comparative analysis of different flow fields at different inclination angles in this section. These studies showed that optimization of blade aerodynamic characteristics, in other words, the shape, could positively impact all aspects of turbocharger characteristics [24]. However, design analysis of the upper blade (that is, the windward side) of the blade is rare when searching [25], which provides ideas for the research strategy and innovation of this paper [26].

In recent years, the first stage blades of each aero-engine turbocharger are bent towards the leading edge and the swept-back. The reason is that this kind of type can effectively reduce the blade fatigue failure rate, and at the same time, the efficiency and the outflow rate are meant to be significantly improved.

The purpose of this paper is to translate and apply these ideas to the impeller of turbochargers. On this basis, a new type of blade is developed. The advantages of the new blade in pressure, stress, and deformation are illustrated by software simulation, and a more suitable blade size is obtained by comparison.

2. Materials and Methods

With the continuous improvement of various design parameters of the compressor, the running conditions and dynamic characteristics of the impeller are becoming more and more complex. Considering only the static stress under centrifugal force, the reliability of the impeller cannot be satisfied. As the core part of the compressor, the interaction between fluid and structure and the unique dynamic behavior of the impeller should be considered in the initial stage of design.

The fluid–structure interaction of turbomachinery is an important branch of fluid–structure interaction. The analysis of the fluid–structure coupling phenomenon in the impeller is of great significance for improving the compressor performance, structural strength, and other design levels, enhancing reliability and safety, and prolonging the working life of the compressor.

Fluid–structure coupling analysis refers to the coupling of fluid analysis and solid analysis, whose research object is the flexible solid [27]. The contents of the study are the various behaviors under the action of the flow field and the influence of the flow field on deformed solids. In the research and analysis of superchargers, the deformation force of high speed and high-pressure flow field on the blade, and the distribution and magnitude of fluid motion and load under different aerodynamic profiles of the blade are involved. Therefore, this paper chooses the method of fluid–structure coupling analysis and uses ANSYS Workbench to conduct CFD analysis on the impeller [28].

2.1. The Turbulence Models

Turbulence is characterized by high complexity, three-dimensional instability, and irregularity with rotation [29,30]. Among them, the numerical simulation of turbulence occupies a very high proportion in the calculation of fluid mechanics. The current numerical calculation methods of turbulence can be summarized as follows: Direct Numerical Simulation (DNS) [31], Large Eddy Simulation (LES), and Reynolds Average Numerical Simulation (RANS). The DNS method is to calculate turbulence directly using instantaneous Navier–Stokes equations. There is no need to simplify or approximate turbulent flow in the calculation process. Theoretically, DNS can obtain relatively high precision results [32], but DNS also has some disadvantages, that is, high requirements on computer performance. When using this method to solve the problem, it needs large memory space and high computing speed. Therefore, at present, it can only be used to simulate turbulent flows with simple geometric and physical boundaries and low Reynolds numbers, or to carry out some exploratory work, but cannot be used for engineering calculation in a real sense. In the LES method, the turbulent large eddy vortices can be directly stimulated by the instantaneous Navier–Stokes equations. The small-scale vortices, however, cannot do the same, further interpreting as they cannot be used in impeller-related engineering research for the time being [33]. To sum up, we mainly adopt the RANS method [34].

Reynolds Average Numerical Simulation

The basic idea of the Reynolds average is the Reynolds hypothesis, which is to express the physical quantity of turbulence by the sum of mean and pulsation value so that the turbulence can be regarded as the superposition of mean and pulsation field. The core of the method is to solve the time-homogeneous Reynolds equation rather than the instantaneous Navier–Stokes equation [35]. The advantage of this method is that it cannot only reduce the amount of calculation but also achieve better results in engineering practice. At present, the Reynolds average method is widely used in the numerical simulation of turbulence [36].

The instantaneous velocity and instantaneous pressure can be decomposed into the sum of the average quantity and the pulsation quantity according to the Reynolds time mean rule, which can be expressed as:

$$u_i=\overline{u_i}+{u^{\prime }}_i^{}$$

(1)

$$p=\overline{p}+p^{\prime }$$

(2)

The Navier–Stokes equations of Reynolds average can be expressed as:

$$\frac{\partial {\overline{u}}_j}{\partial x_j}=0$$

(3)

$$\frac{\partial {\overline{u}}_j}{\partial t}+{\overline{u}}_j\frac{\partial {\overline{u}}_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \overline{p}}{\partial x_j}+\frac{\partial }{\partial x_j}\left(v\frac{\partial {\overline{u}}_i}{\partial x_j}-\overline{{u^{\prime }}_i^{}{u^{\prime }}_j^{}}\right)$$

(4)

where $-\overline{{u^{\prime }}_i^{}{u^{\prime }}_j^{}}$ represents Reynolds’ stress or turbulent stress.

According to the different numerical simulation methods, the turbulence models based on the Reynolds average method can be divided into two categories: Reynolds Stress Model (RSM) and Eddy Viscosity Model (EVM) [37]. The Reynolds Stress Model (RSM) includes Algebraic Reynolds Stress Model (ARSM) and Reynolds Stress Model (RSM) [38]. The Eddy Viscose Model (EVM) includes a zero-equation model, one equation model, and two equations model.

Based on the Reynolds average method, the flow field inside the turbocharger impeller is calculated by using a turbulence model with high order anisotropy. The standard model needs to solve two equations, namely, the turbulent kinetic energy equation and the dissipation rate equation. The turbulent kinetic energy transport equation is derived from the exact equation, but the dissipation rate equation is obtained by physical reasoning and mathematical simulation of similar primitive equations. The flow in the model can be assumed to be completely turbulent and the influence of viscous molecules can be ignored. Therefore, standard models are only suitable for simulating turbulent flow processes.

The dissipation rate of turbulent kinetic energy ε is defined as [39]:

$$\epsilon =\frac{\mu }{\rho }\left(\frac{\partial {u^{\prime }}_i^{}}{\partial x_k}\right)\left(\frac{\partial {u^{\prime }}_i^{}}{\partial x_k}\right)$$

(5)

Turbulent viscosity can be expressed as a function of turbulent kinetic energy k and dissipation rate ε:

$${\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }$$

(6)

The equation of turbulent kinetic energy k and dissipation rate ε of the standard k-ε model can be expressed as follows:

$$\frac{\partial (\rho \epsilon )}{\partial t}+\frac{\partial \left(\rho \epsilon u_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_{1\epsilon }\frac{\epsilon }{k}\left(G_k+C_{3\epsilon }{G}_b\right)-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}+S_{\epsilon }$$

(7)

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial \left(\rho k{u}_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left[\left(\mu +{\mu }_t\right)\frac{\partial k}{\partial x_j}\right]+G_k+G_b-\rho \epsilon -Y_M+S_k$$

(8)

where the model constants are C_1ε = 1.44, C_2ε = 1.92, and C_3ε = 0.09 respectively, and the turbulent Prandtl numbers of turbulent kinetic energy and dissipation rate are σ_k = 1.0, σ_ε = 1.3, respectively [40].

2.2. Arbitrary Lagrangian–Eulerian Equation

To simulate the behavior of a solid medium, the LaGrange motion formula is generally used, for example, tracking the motion of particles in the medium [41]. However, Euler’s formula is usually used to analyze fluid flow to comprehend the motion of fluid at a specific position in space. The fluid domain varies with time when it interacts with a solid or with a free surface, both of which are taken into account. At this point, the Arbitrary Lagrangian–Eulerian equation (ALE) is needed to describe the motion [42]. The coordinates of Lagrange, Euler and ALE formula are shown in Figure 1. ALE equation is a verified and discussed solution combining the Lagrange formula and Euler’s formula.

2.2.1. Governing Equations of Solid

The Lagrange equation of the structure is:

$$\rho \frac{{\partial }^{2}u}{\partial t^2}=\nabla \cdot \tau +f^B$$

(9)

where ρ is the density, u is the structural displacement vector, t is the time, τ is the Cauchy stress tensor, f^B is the volume force vector, and (∇) represents the divergence operator of structural configuration deformation.

The linearity of Equation (9) depends on the constitutive relation and displacement of the selected material

The boundary conditions of Equation (1) need to be solved are:

$$\begin{array}{c}u=u_s on S_u\\ \tau n=f^S on S_f\end{array}$$

(10)

where S_u and S_f represent the boundary part with the specified displacement U_s and traction f^S, and n represents the outward unit normal vector to the boundary.

2.2.2. Governing Equations of Fluid

In ALE motion description, the equation of motion of a compressible Newtonian fluid can be expressed as:

$$\rho \frac{\delta v}{\delta t}+\rho \left[\left(v-\widehat{v}\right)\cdot \nabla \right]v=\nabla \cdot \tau +f^B$$

(11)

$$\frac{\delta \rho }{\delta t}+\left(v-\widehat{v}\right)\cdot \nabla \rho +\rho \nabla \cdot v=0$$

(12)

$$\rho \frac{\partial e}{\delta t}+\rho \left(v-\widehat{v}\right)\cdot \nabla e=\tau \cdot D-\nabla \cdot q+q^B$$

(13)

where ρ refers to the liquid density, $\frac{\delta }{\delta t}$ is the inverse of the total time “seen” by the probe as it moves in the ALE coordinate system, v is the speed of the fluid, τ is the fluid stress tensor, f^B is the fluid volume force vector, e is the specific internal energy, D represents the velocity strain (strain rate) tensor and has $2D=\nabla v+{\left(\nabla v\right)}^T$, q is the heat flux vector, q^B is the rate of heat generation per unit volume, (∇) and (∇) represent the divergence operator and the gradient operator, respectively, and (·) represents the inner product [43].

Equation (11) is the momentum equation, Equation (12) is the mass conservation equation, and Equation (13) is the energy conservation equation.

Notice that in Equations (11)–(13), if $\widehat{v}=0$ which means no shift in the ALE coordinate system (or finite element discretized grid), then Euler’s equation can be assumed. Otherwise, if $\widehat{v}=v$, which means ALE coordinates move with fluid particles, it transforms into a Lagrange equation of motion.

The constitutive relation of Newtonian fluid is:

$$\tau =\left[-p+\lambda \nabla \cdot v\right]I+2uD$$

(14)

where ρ is the fluid pressure, I is the constant tensor, and u and λ are the first and second viscosity coefficients, respectively.

$$\lambda =-\frac{2}{3}u$$

(15)

For most conditions, the Stokes hypothesis (15) can accurately describe the behavior of fluid flow because it is more widely used.

The heat transfer constitutive equation is established:

$$q=-k\nabla \theta$$

(16)

where k is the conductivity tensor (reduced to a separate value in isotropic media) that presents the temperature.

We also need equations of state to solve Equations (11)–(13).

$$\rho =\rho \left(p,\theta \right)$$

(17)

$$e=e\left(p,\theta \right)$$

(18)

Notice that density (fluid particles) is not a function of time in incompressible fluids, and $\nabla \cdot v=0$ in (12). On this basis, Equations (11) and (12) are sufficient to solve the isothermal motion of an incompressible fluid, so energy Equation (13) need not be considered; but when dealing with compressible fluids, Equations (11)–(13) require all the calculations [44].

When the fluid viscosity can be ignored in the model, the Euler equation of motion is used. Because the conditions of supersonic/subsonic flow and viscous/inviscid flow must be considered, the boundary conditions required in the most general case of solving Equations (11)–(13) are beyond the scope of this paper and will not be discussed temporarily. When the ale equation is used to calculate the fluid flow equation, factors such as boundary movement, free surface, and fluid–fluid interaction can be considered.

Under dynamic boundary conditions, the following conditions must be met:

$$\begin{array}{c}\widehat{u}·n={\widehat{u}}_s on S_{\widehat{u}}\\ \widehat{u}·t={\widehat{u}}_{t } on S_{\ddot{u}}\end{array}$$

(19)

where S_ȗ is the surface part corresponding to the displacement ȗ_s and ȗ_t applied in the normal and tangential directions, n and t are the unit normal vector and unit tangent vector of the boundary, respectively, and ȗ is the boundary displacement. If the fluid model is a non-viscous fluid, the second equation in Equation (19) does not apply.

When considering flow–flow interaction, compatibility and equilibrium conditions must be met at the interaction interface. The compatibility condition ensures that the particle velocity of the two fluids at the interface is the same (no-slip condition). If both fluids are viscous, the normal component of the velocity of the interaction between the two fluids is equal (slip condition) [45].

The equilibrium conditions of flow–flow interaction are:

$$\left({\tau }_2-{\tau }_1\right)·n=\alpha \left(\frac{1}{R_1}+\frac{1}{R_2}\right)n$$

(20)

In the above formula, τ₁ and τ₂ are the stress tensors corresponding to two interacting fluids, n is the unit normal vector of the action interface pointing outward from surface 1, α is the surface tension coefficient between fluids, and R₁ and R₂ are the principal radii of curvature of the interaction surface (if the curvature center is on one side of fluid 1, it is defined as positive, otherwise it is negative). If the surface tension effect is ignored, $\alpha$ = 0 in Equation (20).

If free surfaces are considered, Equation (20) still applies. Among them, the influence of a fluid (generally air) is usually only the pressure $p_0$ (assuming that the fluid is inviscid), formula 20 can be replaced by:

$$-p_{0}n-\tau ·n=\alpha \left(\frac{1}{R_1{}^{\prime }}+\frac{1}{R_2}\right)n$$

(21)

Here, n points outward from the free surface. If their center point is on one side of the model fluid, the radius of curvature is assumed to be positive.

Since only one fluid has been explicitly considered in the model so far, Equation (21) is not enough to describe the motion of a free-form surface, so an equation is needed.

Suppose the surface function $S\left({}^{0}x,t_0\right)=0$ at the reference time t₀ where ${}^{0}x$ is the coordinate vector of the particle on the free surface at a time t₀, and the following conditions are met:

$$\frac{\delta S}{\delta t}+\left(v-\widehat{v}\right)\cdot \nabla S=0$$

(22)

It ensures that at the time t₀, the particles on the free surface will stay on the surface forever [46].

2.2.3. Fluid–Solid Interaction Equations

In the interaction between viscous fluid and solid medium, the equilibrium and compatibility conditions should also be satisfied at the fluid–solid interface, so:

$${\tau }^S\cdot n={\tau }^F\cdot n$$

(23)

$$u^I\left(t\right)={\widehat{u}}^I\left(t\right)$$

(24)

$${\dot{u}}^I\left(t\right)=v^I\left(t\right)={\widehat{v}}^I\left(t\right)$$

(25)

$${\ddot{u}}^I\left(t\right)={\dot{v}}^I\left(t\right)={\dot{\widehat{v}}}^I\left(t\right)$$

(26)

where n indicates the unit vector perpendicular to the fluid–solid interface, u and ȗ indicate the displacement of the structure and fluid domain (or grid in finite element analysis), v indicates the fluid speed and $\widehat{v}$ indicates the speed of the fluid domain. The dot indicates the time derivative, and I, S, and F indicate the interface, solid media, and fluid media.

2.3. Fluid–Structure Interaction Model

2.3.1. Equations of Fluid Motion and Boundary Conditions

According to the theory of small disturbance potential, the equation representing the fluid motion law can be derived, as shown below:

$${\nabla }^{2}p-\frac{1}{a^2}p=0$$

(27)

The dynamic pressure field in the fluid region is approximated by a linear combination of basic functions:

$$P(x\cdot y\cdot z,t)=N(x\cdot y\cdot z)\cdot P(t)=\sum _{j=1}^H{N}_j(x\cdot y\cdot z)P_j$$

(28)

Further discretized in space into ordinary differential equations:

$$\sum _{j=1}^H\left[\frac{1}{a^2}{{\displaystyle \int }}_j\frac{1}{g}{N}_i{N}_{j}ds\right]P_j+\sum _{j=1}^H\left[{{\displaystyle \int }}_v\nabla N_i\nabla N_{j}dv\right]P_j={{\displaystyle \int }}_j{N}_i\frac{\partial P}{\partial n}ds$$

(29)

The wave equation is:

$$\left[M^L\right]\left\{P\right\}+[k]\left\{P\right\}=\left\{f_L\right\}$$

(30)

In the equation above, we have:

$$\left\{P\right\}=\left\{\begin{array}{l}{P}_J^L\hfill \\ P_e^L\hfill \end{array}\right\}$$

(31)

2.3.2. Discrete Model of Solid Motion Equation

The discrete model of the solid motion equation is:

$$\left[\begin{array}{cc}{M}_{JJ}^g& M_{Je}^g\\ M_{eJ}^g& M_{ee}^g\end{array}\right]\left\{\begin{array}{l}{\overline{U}}_J^g\hfill \\ {\overline{U}}_e^g\hfill \end{array}\right\}+\left[\begin{array}{cc}{K}_{JJ}^g& K_{Je}^g\\ K_{eJ}^g& K_{ee}^g\end{array}\right]\left\{\begin{array}{c}{U}_J^g\\ U_e^g\end{array}\right\}=\left(Q\right)+\left\{\begin{array}{c}{f}_J\\ 0\end{array}\right\}$$

(32)

In the equation above:

$$\left\{U^g\right\}=\left\{\begin{array}{c}{U}_J^g\\ U_e^g\end{array}\right\}$$

(33)

The pressure vector at the fluid interface is $\left\{f_j\right\}$.

2.3.3. Motion Equation of the Coupled System

The motion equation of the coupled system is:

$$(\left[\begin{array}{cc}\left[M^L\right]& \rho {\left[T_J^L\right]}^T\left[T_J^g\right]\\ 0& M^g\end{array}\right)\left\{\begin{array}{c}\ddot{P}\\ {\ddot{U}}_g\end{array}\right\}+\left(\begin{array}{cc}\left[K^L\right]& 0\\ -\left[T_J^L\right]& \left[K_g\right]\end{array}\right]\left\{\begin{array}{c}p\\ U^g\end{array}\right\}=\left\{Q\right\}$$

(34)

2.4. Solving Method

According to the different coupling degrees of the physical field between fluid domain and solid domain, fluid–structure coupling problem analysis can be divided into the strong coupling and weak coupling. The corresponding solution methods are the direct solving method and separation solution method, respectively. The direct solution is to couple the governing equations of the flow field and structure field into the same equation matrix, that is, to solve the fluid–solid governing equations simultaneously in the same solver [47]. It is very advanced in theory and is suitable for large solid deformation, biological diaphragm movement, etc. Yet, in practical application, it is difficult to combine the existing computational fluid dynamics and computational solid mechanics technology with the direct method [48].

In addition, considering the convergence difficulty and time-consuming problem of the synchronous solution, the direct solution method is mainly applied to the simulation analysis of simple problems such as thermal–structure coupling and electromagnetic-structure coupling. Only some very simple studies have been done on fluid–structure coupling, and it is difficult to be applied to practical engineering problems. The separation method of weak fluid–structure coupling is to solve the governing equations of fluid and solid respectively and to transfer data through the fluid–structure coupling interface [49]. This method can be used to solve practical large-scale problems because of its reduced demand for computer performance.

To sum up, in software simulation, fluid–structure coupling analysis adopts the separation method. The fluid–structure interaction analysis algorithm and function in ANSYS are quite mature. Fluid–structure interaction analysis of ANSYS Mechanical APDL + CFX, ANSYS Mechanical APDL + FLUENT, ANSYS Mechanical + CFX can be implemented with or without third-party software (such as MPCCI).

2.5. The Specifics of the Impeller

The original impeller model is an A-type turbocharger for a diesel engine developed by the D factory. The compressor impeller is the radial straight blade; impeller diameter is 315 mm. The guide vane adopts a three-time parabolic structure with long and short blades. The guide vane and the compressor impeller are pressed together on the bushing to form an integral whole. The impeller adopts advanced long and short vanes technology, which can improve the pressure ratio and efficiency.

In the test of the 16VD240ZJD diesel engine with this type of supercharger, the following results are obtained at 50, 80 and 100% calibrated power conditions in

Table 1. Diesel engine test under various conditions.

	Condition 1	Condition 2	Condition 3
Engine speed/power (rpm)/kW	840/1480 (50%)	960/2600 (80%)	1000/3200 (100%)
Main exhaust (°C)	452	465	525
Boost pressure	60	166	227

, respectively:

The maximum speed of the compressor impeller is 27,500 rpm under full power conditions. In this discussion, working condition 2, namely, 80% calibrated power working condition, and compressor speed 22,000 rpm was selected.

The impeller blade is made of K418 casting alloy, which is γ′ phase precipitation-strengthened nickel-base casting alloy. It has good embedding strength, thermal fatigue, and oxidation resistance under 900 °C. The material density is ρ = 8.0 g/cm³. On this very note, the elastic modulus is E = 195 Gpa, and Poisson’s ratio is σ = 0.25 [50].

Blade configuration is shown in Figure 2:

As shown in Figure 2, we have drawn the initial model of a turbocharger impeller by AutoCAD. A is the top view, C is the side view, and B is the top view of a single long blade.

This initial model is hereinafter referred to as type O.

2.5.1. Model of Type A Blade

Based on the original blade length, we changed the shape of the windward side on the premise of keeping the diameter length of the blade chord unchanged at 60 mm. In AutoCAD, we curve stretch the windward contour, and the new contour is shown in the yellow curve in Figure 3. The leading edge is protruded and bent and a swept-back design is carried out in the second half as the chord length unfolds. The swept part is designed with an arc to add an inlet angle to the blade. The chord inclination angle of the given front segment (inside) is 20°. The backward displacement distance α = 8 mm between the starting point of the front segment and the shaft sleeve and the original starting point of the o-blade root.

The design size is shown in Figure 3 and Figure 4:

2.5.2. Model of Type B Blade

Similarly, based on type O, the design idea is the same as it is in type A. The difference is that α = 2 mm is set at the same time as the inclination angle of the front (inner) string is 20°. The design size of Type B is shown in Figure 5 and Figure 6.

3. Results and Discussion

The type O/A/B models are established, respectively. The simulation type of FLUENT in ANSYS is a steady-state simulation, adopting the k-ε turbulence model. The working medium of the fluid domain is air and the fluid–structure interaction is calculated and analyzed. K418 casting alloy was selected as the solid structure material. There are two kinds of impeller load: one is inertia force, the other is aerodynamic force, in which inertia force includes centrifugal force caused by impeller rotation and impeller’s gravity and aerodynamic force is the force generated by the fluid acting on the fluid–structure interaction interface in a compressor. In the inertial load option, the rotation speed is 22,000 rpm under a given working condition, and the rotation direction is counterclockwise around the impeller spindle (as seen from the top view). Gravity has been excluded from this calculation.

The flow field was set for the type O/A/B model, and fluid–structure coupling was calculated for the three models in the Workbench.

3.1. Fluid Analysis

3.1.1. Pressure Distribution

As can be seen from Figure 7, in the O-type model, the pressure on the lower edge of the blade near the wheel and the windward edge of the upper edge is higher, while the pressure on the curved surface in the middle of the blade and near the outer edge is less. The windward side of the maximum pressure reached 12,021.98 Pa. At the lower edge of the blade near the wheel, due to casting conditions and its integrity, the structural strength can ensure that the blade is easily capable to withstand great pressure without deformation and vibration. For the windward side of the upper edge of the blade, the relative degree of freedom constraints is less, and this part is most prone to deformation. Therefore, higher fluid pressure will lead to blade vibration, surge, and fracture.

In the A-type model in Figure 8, the pressure distribution is the same as that of the O-type. The leading edge of the improved curved windward edge is subjected to large gas pressure, with a maximum value of 11,652.50 Pa which is reduced relative to O-type. As shown in Figure 9, when the α in the blade parameter is changed to 2 mm, that is the type B model. In this case, the pressure distribution of the whole blade subjected to fluid becomes more uniform compared with the first two cases. The part of the blade subjected to maximum pressure is reduced, and the pressure gradient on the upper edge of the blade which is most prone to failure is relatively mild as well. The pressure on the curved leading edge which is easy to fracture is 6482.76 Pa which is a significant reduction compared to O-type and A-type. In terms of fluid pressure, the swept-back blade design with a curved leading edge has obvious advantages, and the gas pressure is softer and more uniform at work. Among them, the shape of α = 2 mm has a more obvious advantage than that of α = 8 mm. This is roughly the same as the experimental results of Zhao et al. [51].

In Figure 7, Figure 8 and Figure 9, the numbers on the left are relative to the operating pressure. In the simulation process of Fluent, we define the pressure exerted by the fluid on the structure surface as the positive pressure. Under the current boundary conditions, the default operating pressure in the software is 101,325 Pa. The negative value on the left of the figure represents the value where the pressure is less than the operating pressure. The parts below −101,325 Pa indicate the absolute negative pressure relative to the surface pressure. In Figure 7, the minimum value of the pressure distribution is −122,268.99 Pa, that is, the minimum absolute negative pressure in the flow field is −20,943.99 Pa. For the same reason, in Figure 8 and Figure 9, the minimum absolute negative pressure is −31,726.19 and −12,284.8 Pa, respectively. In these three models, the minimum negative pressures are mainly distributed at the ridge of the large blades and the tip of the small blades. Even so, B-type is much better than O-type for the distribution of the negative pressure.

3.1.2. Outflow Rate

The gas outflow rate of the turbocharger impeller under three conditions is obtained, as shown in the following table.

From

Table 2. The outflow rate of three models.

	Type O	Type A	Type B
Outflow rate(m3/s)	5.4655	5.7725	5.7354

, it can be intuitively seen that the gas outflow flow was obtained at the same speed of 22,000 rpm under the flow field of three types. Compared with O-type, the outflow rate obtained by A-type increases significantly. The gas flow rate obtained in B-type is 5.7354 m³/s, slightly lower than A-type, but still shows an increasing trend compared with O-type.

Table 2 shows that the swept-back blade with a curved leading edge can effectively increase the outflow rate of the turbocharger compressor compared with the traditional blade. The inlet pressure of the turbocharger takes the standard atmospheric pressure. At the same speed, the higher the flow rate, the heavier the compressor outlet pressure, so the higher the pressure ratio. The compressor pressure ratio is an important index to measure the performance of a turbocharger. Therefore, from the perspective of outflow rate, the improved blade can enhance the compressor pressure ratio considerably.

3.2. Structural Analysis

In the structural module, the impeller is analyzed by the finite element method. Considering the effect of centrifugal force and initial load, the flow field results obtained in 3.1 was imported into the static structure module for analysis. As the surface load of the impeller, it is loaded into the stress analysis model. On this basis, the working speed of the impeller was set at 22,000 rpm, and the simulation results under fluid–structure interaction were obtained.

3.2.1. Equivalent Stress Distribution

It can be seen from the results of the equivalent force analysis of the impeller in Figure 10, Figure 11 and Figure 12 that when the impeller rotates at high speed, the equivalent force distribution is mainly in the upper edge of the blade and the lower part of the blade near the outer side, and the maximum stress is located in the upper end of the blade near its root.

In Figure 10, the maximum stress point of the O-type model is at the windward side near the blade root, where the average stress is 725.53 MPa, and the highest point equivalent stress even reaches 816.1 MPa. In Figure 11, in the blade with α = 8 mm, the difference of stress on the middle and lower lateral sides of the blade is alleviated. The average value of the maximum stress at the upper end of the blade is 1057.7 MPa, and the maximum value is 1612.9 MPa. However, the maximum stress is closer to the root of the blade or even at the junction of the blade and disc. The structural strength of this part is extremely high, which is far from that of the most stressed part in O-type and the part prone to deformation fracture. As can be seen from Figure 12, in B-type with α = 2 mm, the equivalent stress distribution is similar to A-type, and the stress distribution on the whole impeller is more even. The maximum equivalent stress is reduced to 111.99 MPa at the point of maximum stress close to the root of the blade. The metal fatigue strength of the compressor impeller blade is determined by the stress and the structural strength of each part in the high-speed rotation. The vibration and deformation of blades are even affected directly. In the case of the same structural strength, the less stress, the less prone to failure.

Therefore, it can be concluded from Figure 10 to Figure 12 that the swept-back blade with a curved leading edge has an obvious advantage in terms of equivalent stress, which can significantly reduce the overall stress in the blade in the flow field. This is roughly the same as the experimental results of Danilishin et al. [52].

3.2.2. Total Deformation

Figure 13, Figure 14 and Figure 15 suggest that when the impeller rotates at high speed, the deformation caused by fluid–structure coupling growingly increases outward from the root of the blade. In the three models, the largest part of deformation is located from the upper edge of the blade to the edge corner, and the maximum displacement in O-type is 0.80565 mm. As shown in Figure 14, the inlet bend Angle is designed on the outer upper end of the type A-shaped blade so that the structural area of this part is sharply smaller than that of the middle part, which results in the maximum deformation displacement of A-type reaching 2.8486 mm during fluid–structure coupling deformation. In this context, it is greatly increased compared with O-type. Nevertheless, it can be seen from Figure 15 that when α = 2 mm is utilized for B-type blades, not only the maximum deformation position is improved but also the maximum deformation displacement is reduced to 0.11733 mm compared with the first two cases, and the distribution is more uniform at the leading edge and swept-back corner. Not only is the condition of a = 8 mm sharply reduced, but it is also obviously improved compared with the original model O-type. Compressor blade deformation and displacement are generally prone to failure areas. The smaller the deformation under fluid–structure interaction, the smaller the probability of vibration and even surge or fracture. It can be concluded by comparing Figure 13, Figure 14 and Figure 15 that the new swept-back blade with a curved leading edge can considerably improve the blade deformation in the flow field and reduce the failure rate under appropriate shape parameters (such as a = 2 mm), which is roughly the same as the experimental results of Saravanan et al. [53].

3.3. Discussion

Usually, the cause of impeller blade fracture is basically the same, which belongs to fatigue fracture. In fact, the crack is about one-third to two-thirds of the total length of the windward side, as shown in the following Figure 16.

This paper discussed the main design scheme, the main idea is to optimize the shape and windward Angle, so as to achieve the strength of vulnerable parts, and reduce the possibility of failure. The design basically achieved the purpose through simulation results. In the new configuration, especially the B-Type, the protruding leading edge design effectively enhances the rigid strength of the fracture-prone area on the windward side. Due to the design reasons, the pressure and equivalent stress in the fracture-prone area are obviously reduced, and the possibility of rigid fracture of the blade is reduced. At the same time, the new shape greatly reduces the maximum displacement of blade vibration and reduces the incidence of fault caused by vibration. On the whole, this modeling optimization can significantly improve the turbocharger operation process, effectively improve its life, and improve the flow rate, enhance the overall performance of the compressor.

4. Gray Correlation Analysis

There are many kinds of multivariate analysis methods in statistics, among which fuzzy cluster analysis (FCA) is one of the commonly used methods [54]. By using mathematics to analyze the correlation degree of uncertainty or fuzzy things, we can objectively classify the research objects. Grey relational analysis (GRA) is a research method to obtain information through comparative research. Its biggest feature is that it can calculate the influence degree of various factors. The optimization model of this paper is proposed based on GRA theory, and the influence of various factors on outflow is analyzed by using this model [55]. The main steps are as follows:

(1)

Determination of sequence

The reference sequence y and the comparison sequence x together form a grey correlation sequence. The sequence of system behavior characteristics is generally composed of one or more reference sequences, which are composed of different groups of statistical data. The development trend of the system can be perfectly displayed by these reflections. The reference sequence is represented as follows:

$$Y_t=\left[y_t\left(1\right)y_t\left(2\right)y_t\left(3\right)\dots y_t\left(n\right)\right]$$

(35)

where Y_t represents the t-th reference sequence; (t = 1, 2, …, n), [y_t(1), y_t(2), …, y_t(n)] are the data of the reference sequence in group Y_t. The performance of the Y_t reference sequence variation law can be perfectly reflected by the above data

If the system is affected by at least m factors and n working conditions, the eigenvector matrix of its influencing factors can be obtained, and the results are as follows:

$$X_t=\left[\begin{array}{l}{x}_{t1}\\ x_{t2}\\ \dots \\ x_m\end{array}\right]=\left[\begin{array}{llll}{x}_{t1}\left(1\right)& x_{t1}\left(2\right)& \dots & x_{t1}\left(n\right)\\ x_{t2}\left(1\right)& x_{t2}\left(2\right)& \dots & x_{t2}\left(n\right)\\ \dots & \dots & \dots & \\ x_{tm}\left(1\right)& x_{tm}\left(2\right)& \dots & x_{tm}\left(n\right)\end{array}\right]$$

(36)

where X_t is the comparison sequence corresponding to reference sequence Y_t. From this comparison sequence, there are m influencing factor vectors and the j-th one is x_tj, (j = 1, 2,..., m).

In this study, the reference sequence Y represents outflow. Comparison sequence X₁–X₃ represents press, equivalent stress, and total deformation, respectively. The sequence matrix of each factor measured in this experiment is as follows:

$$\left[\begin{array}{l}Y\\ X1\\ X2\\ X3\end{array}\right]=\left[\begin{array}{lllll}5.7725& 5.7703& 5.7637& 5.7459& 5.7354\\ 11652.5& 11244.1& 9367.91& 7557.69& 6482.76\\ 1612.9& 1350.7& 943.15& 429.63& 111.99\\ 2.8486& 2.0747& 1.6159& 0.53185& 0.11733\end{array}\right]$$

(37)

(2)

Dimensionless original sequence

Through the analysis, it can be seen that different research factors represent the corresponding dimensional meaning and physical meaning. Therefore, to improve the accuracy of the test and reduce unnecessary errors caused by calculation, the parameter characteristics of different sizes should be dimensionless, and the interval dimensionless transformation sequence is adopted:

$$x\left(k\right)=\frac{x\left(k\right)-\min x\left(k\right)}{\max x\left(k\right)-\min x\left(k\right)};k=1,2,3\dots ,n$$

(38)

To make the calculation more accurate and simplify the calculation steps, MATLAB software will be used for the calculation. Figure 17 shows the fuzzy membership grade (FMG), Euclidean grey relational grade (EGRG), and fuzzy grey relational grade (FGRG) of three structural factors on outflow. Among them, the fuzzy membership grade, Euclidean grey relational grade, and fuzzy grey relational grade of pressure discharge are the highest. The higher the fuzzy memory level, the better the similarity between the research factors. Fuzzy grey relational grade of pressure, equivalent stress, and total deformation are 0.8596, 0.6131, and 0.8001, respectively. Therefore, the pressure has the greatest impact on the outflow, followed by the total deformation, and the effect of equivalent stress is relatively small.

5. Conclusions

The structural faults of the turbocharger impeller are mainly caused by the exciting force, which seriously affects the life and working conditions of the turbocharger [56,57,58,59]. Blade shape optimization is an effective means to reduce the failure rate and improve the efficiency of the compressor. A swept-back blade with a curved leading edge and intake Angle was introduced into the design of the turbocharger blade. This optimization design is the first time that this kind of shaped blade is applied to the compressor impeller of a small turbocharge. Fluid–structure coupling calculation was carried out for the new blade and the original blade at 22,000 rpm, and the calculation results were compared and analyzed. The main conclusions are as follows.

(1)

In the flow field, the swept-back blade to pressure distribution is more uniform than that of the original blade. In the windward part with the maximum pressure, the pressure of the new blade is significantly decreased, and especially the blade design with α = 2 mm has obvious advantages.

(2)

At the same speed and inlet pressure, swept-back blades can obtain more flow to the outlet, so they can obtain a greater compressor pressure ratio.

(3)

In the impeller structure, the maximum equal effect force point of the swept blade is closer to the blade root with strong structural strength, far away from the leading edge, inlet Angle with the largest deformation displacement, and prone to fracture. In particular, the blade with α = 2 mm can not only greatly reduce the stress on the blade in the flow field, but also significantly improve the blade displacement to deformation.

(4)

The fuzzy grey correlation degrees of pressure, equivalent stress, and total deformation concerning flow are 0.8596, 0.6131 and 0.8001, respectively; the biggest relationship with outflow is the pressure, followed by the total deformation, and finally, the equivalent pressure.

According to the data analysis, the impeller composed of swept-back blades with a curved leading edge can improve blade motion in almost all aspects, reduce stress and deformation, and improve the pressure ratio at high speed. Among the three known designs, the best option is the blade with α = 2 mm, which minimizes the probability of failure and increases the pressure ratio, thereby improving the turbocharger’s working life and efficiency. Swept-back design of blade shape can be more widely applied in the field of the turbocharger, which is of positive significance to further optimize the compressor.