Turbocharger containment is the capability of a turbine or compressor housing to contain a burst wheel. It is essential to ensure the product safety and to prevent.
Explicit dynamic finite element simulation of turbocharger containment and wheel burst L. Wang, D. M. Eastwood Applied Mechanics, Cummins Turbo Technologies, Huddersfield, UK. ABSTRACT
Turbocharger containment is the capability of a turbine or compressor housing to contain a burst wheel. It is essential to ensure the product safety and to prevent catastrophic accidents. In recent years, Cummins Turbo Technologies developed an explicit Finite Element Analysis (FEA) based approach to simulate the turbocharger containment, aiming to minimise the cost associated with testing, optimise the product design and shorten the new concept development cycle. In this study, a strain rate and temperature dependent plasticity material model with fracture modelling capability is employed to model the turbine housing material behaviour. In addition, a methodology of optimising the weakening slot design to burst a turbine wheel at any specified speed is presented.
Keywords: Explicit Dynamic FEA, ANSYS Autodyn, turbocharger, containment, wheel burst
NOMENCLATURE
A
Initial yield stress
C
Strain rate constant
B I
M 𝑚𝑚 n
Hardening constant
Internal element force Mass matrix Thermal softening exponent Hardening exponent
P T
Current temperature
u
Displacement
Δt
Time increment
u
Velocity
u
Acceleration
TH
Tmelt
Troom
εp ε∗p σ
1.
External applied force
Homologous temperature Melting temperature Room temperature Effective plastic strain Plastic strain rate Stress
INTRODUCTION
Turbocharger containment is the capability of a turbine or compressor housing to contain a burst wheel. Since the blade tip speed of a wheel can reach 600 m/s, it is critical to ensure the product safety and to prevent catastrophic accidents. In the turbocharger industry experimental testing has been widely used to investigate turbocharger containment, due to its “straightforward” nature and its capability to take material variation as well as geometrical variation into account. Physical testing inevitably costs a significant amount of time and money, however. In recent years, Cummins Turbo Technologies has developed a Finite Element Analysis (FEA) based approach to simulate turbocharger containment, aiming to minimise the cost associated with testing, optimise the product design and shorten the new concept development cycle.
Until now, few papers were published on the topic of numerical simulation of turbocharger containment. Memhard et al [1] and Zheng Jing [2] have done a significant amount of work on material testing, aiming to develop the material plasticity and failure models at various strain rates and temperatures for the turbine housing materials. Very limited work was presented, however, regarding the simulation techniques of turbocharger containment. Winter et al [3] did some pioneering work on the FEA simulation of compressor containment, where relatively good correlation was shown between FEA and testing. Two of the major drawbacks of this paper were: firstly no failure criteria were used on the housing material, hence it may be very challenging to quantify the failure of a housing; secondly, the impeller
was built as three parts to represent the three fragments in testing. This may result in difficulty of using this FEA model to predict the burst speed and modelling crack propagation. Most recently, Ramamoorthy et al [4] developed a method to predict the wheel burst speed and an impressive correlation (2% in speed) to test was achieved. However, their criterion to verify the housing containment may be limited.
In this paper, the explicit FEA solution method has been chosen to simulate the containment process, because it is more robust and efficient in modelling problems that involve high nonlinearities, high speed impact and material fracture. A strain rate and temperature dependent plasticity material model with fracture modelling capability is employed to model the turbine housing material behaviour. In addition, a methodology of optimising the weakening slot design to burst a turbine wheel at any specified speed is presented. This approach has been implemented in Cummins Turbo Technologies and good correlation is achieved.
2.
FINITE ELEMENT SOLUTION METHOD
Finite Element solution methods are generally resolved into the implicit method and the explicit method [5]. The implicit FEA method iterates to find the approximate static equilibrium at the end of each load increment. It controls the increment by a convergence criterion throughout the simulation. For a highly non-linear problem, a large number of iterations have to be carried out before finding the equilibrium. Thus the global stiffness matrix has to be assembled and inverted many times during the analysis. Therefore, the computation is extremely expensive and memory requirements are also very high. It is difficult to predict how long it will take to solve the problem or even if convergence can be achieved. Thus the implicit method is preferable to analyse problems under static and simple loading conditions.
The explicit method determines a solution by advancing the kinematic state from one time increment to the next, without iteration. The explicit solution method uses a diagonal mass matrix to solve for the accelerations and there are no convergence checks. Therefore it is more robust and efficient for complicated problems, such as dynamic events, nonlinear behaviours, and complex contact conditions. However, in order to obtain accurate results, the time increment has to be extremely small, which ensures that the acceleration through the time increment is nearly constant. Therefore it typically requires many thousands of increments. At the beginning of the time increment (t), based on the dynamic equilibrium equation [6-7]:
P − I = Mu
(1)
) are calculated as: The nodal accelerations ( u u | (t ) = ( M ) −1 ( P − I ) | (t )
(2)
where M is the nodal mass matrix, P is the vector of externally applied force and I is the vector of internally induced element force. The acceleration of any node is completely determined by the mass and the net force acting on it. Through time the accelerations are integrated using the central difference rule, by which the change of
the velocity ( u ) is calculated from equation (3), assuming that the acceleration is constant:
u | (t + ∆t / 2 ) = u | (t − ∆t / 2 ) +
∆t (t + ∆t ) + ∆t ( ∆t ) 2
u | (t )
(3)
The velocities are integrated through time and added to the displacement (u) at the beginning of the increment to calculate the displacements at the end of the increment:
u | ( t + ∆t ) = u | (t ) + ∆t | ( t + ∆t ) u | ( t + ∆t / 2 )
(4)
The element strain increments dε is calculated from the strain rate, and then the stresses are obtained from the constitutive equations:
σ (t +∆t ) = f (σ (t ) , dε )
(5)
In this study, the Explicit FEA solution method provided by commercial FEA software – ANSYS Autodyn has been chosen to analyse the wheel burst and housing containment of turbochargers.
3.
SIMULATION OF TURBINE WHEEL BURST
In the turbine containment testing there are two failure modes of the turbine wheels, i.e. hub burst and blade detachment [8-9]. To provoke the blade detachment failure, it is required to cut a weakening slot (either angular or straight) on the backface of a turbine wheel [9]. However, trial-and-error is sometimes required in designing the correct weakening slot, to burst the wheel at its target burst speed. Testing inevitably costs a considerable amount of money and takes time. The objective of this study is to develop an analysis-led-testing method by optimising the weakening slot design.
In the FEA models, it is assumed that 1) 2) 3) 4)
Aerodynamics load has no effect on wheel failure and housing containment. The turbine wheel and housing have uniform density and precise geometry. Materials are isotropic, homogeneous and defect-free. Material failure is independent with respect to the types of loadings, strain rate and temperature.
An angular velocity of 218750 rpm, which is the target burst speed of this wheel as per specification of [9], is applied to the wheel only FEA model. Due to the complexity of the geometry tetrahedral elements (first order, explicit) are used to mesh the turbine wheel resulting in 1.8 million elements, as shown in Figure 1.
Weakening slot Figure 1 Meshing of the weakened turbine wheel FE model
The elastic-plastic properties of Inconel 713C, taken from a uniaxial tensile test, are used to simulate the material behaviour of the turbine wheel under the high centrifugal force. A plastic strain based material failure criterion is applied to model the ductile material failure of the wheel burst. It is essential that the failure criterion does not over-predict or under-predict the burst speed for a specified weakening slot design. The failure criterion is achieved by employing the workflow as shown in Figure 2. The comparison between test and simulation is shown in Figure 3.
Figure 2 Workflow of achieving the material failure criterion
(a) Test part of burst speed (b) FE model of burst speed (c) FE model of 98% burst speed Figure 3 Comparison between test and simulation of wheel burst
4.
RESPONSE SURFACE ANALYSIS OF WEAKENING SLOT DESIGN
Response Surface Method (RSM) is a collection of mathematical and statistical techniques for modelling and analysing problems where several variables (input factors) influence the response (output factors) and the objective is to optimise the response [10]. A number of input factors, as shown in Table 1 and Figure 4, could affect the turbine wheel burst process. In this study, a RSM analysis of the wheel burst FEA with two factors, i.e. slot depth and slot Outer Diameter (OD), and three levels (coded as -1, 0 and 1) has been carried out. The lower and upper limits of the inputs vary by approximately 10% from the baseline. For all the FEA runs, the speed is set as the target burst speed of this wheel, the slot angle and slot width remain the same.
Table 1 Input factors and levels in RSM analysis of wheel burst FEA
Figure 4 Geometrical factors of weakening slot
The Central Composite Design (Face Centred) method [10] is applied to generate the RSM runs, as shown in Table 2. The nominal maximum plastic strain, which determines whether wheel burst takes place, is studied as the output of this RSM analysis. If the nominal plastic strain is equal or greater than 1, then wheel burst (material failure) occurs.
Table 2 RSM runs of wheel burst FEA
The statistics software Minitab has been used to analyse the effects of inputs to the outputs. According to the 3D scatterplot and main effects plot, as shown in Figure 5 and 6, high plastic strain (wheel burst) tends to occur if increasing the OD (outer diameter) or increasing depth of the weakening slot. Furthermore, in this study no interaction between slot depth and slot OD exists, as shown in Figure 7.
Figure 5 3D scatterplot of nominal plastic strain vs slot depth and slot OD
Figure 6 Main effects plot for nominal plastic strain
Figure 7 Interaction plot for nominal plastic strain
In the process of optimising the weakening slot, it is essential to find a weakening slot design which can burst the wheel at exactly the target burst speed. It is believed that if the OD or the depth of the slot is set to be too large, the wheel would burst at a speed lower than the target speed and vice versa. Figure 8 illustrates a workflow of optimising the slot depth for a given slot OD and target burst speed. Based on the capability of the actual cutting tool, a minimum slot depth increment, i.e. x, should be used, aiming to find the “boundary” between burst and burst-free of a turbine wheel.
Figure 8 Workflow of optimising slot depth in wheel burst FEA
5.
SIMULATION OF TURBINE HOUSING CONTAINMENT
Containment is the capability of a turbine or compressor housing to contain a burst wheel. It is critical to ensure the product safety and to prevent catastrophic accidents. In Cummins Turbo Technologies, all new products must pass the containment testing before being released to the market. To assist the new concept development, shorten the design cycles and decrease the testing cost, an explicit FEA based technique of housing containment has been developed, by virtually evaluating and comparing various housing concepts before carrying out any physical testing.
The simulation process includes two sequential analyses: 1) Steady-state thermal analysis, where the temperature distribution of components is analysed and used to take the material thermal softening effects into account in 2) Transient explicit dynamic analysis, in which wheel burst as well as housing containment are simulated. The wheel geometry with an optimised weakening slot and the turbine housing are assembled to build the housing containment FEA model. The mesh of which is shown in Figure 9, where 5.1 million elements are used.
The target burst speed is applied to the weakened wheel as the initial condition of the FEA model. Impact from wheel fragments to the housing is modelled by using the penalty formulation of the trajectory contact approach [11], where a contact event is detected, if the trajectory of a node and a face intersects during one computation cycle. A local penalty force is calculated to push the node back to the face. Equal and opposite forces are calculated on the nodes of the face in order to
conserve linear and angular momentum. Frictional effects are considered and a frictional coefficient of 0.2 is assumed between the wheel and housing.
Figure 9 Meshing of housing containment FEA model
In this study, the reduction of elastic modulus due to elevated temperature is neglected because plastic deformation and material fracture dominate in the containment process. The Johnson Cook strength model [12] is used to model the plasticity behaviour of the ductile SiMo iron housing, which is subjected to large strain, high strain rate and high temperature. The yield stress is defined as:
n
where
εp
Y = �A + B�εp � ��1 + Cln�ε∗p ��[1 − (TH )m ]
is the effective plastic strain,
ε∗p
is the plastic strain rate;
homologous temperature, which is defined by
TH = T
T−Troom
melt −Troom
(6)
TH
is the
(7)
A is the initial yield B is the hardening constant, n is the hardening exponent, C is the strain rate constant, and 𝑚𝑚 is thermal softening exponent. There are five material constants in the Johnson Cook model:
stress,
To simulate the material failure there are a number of material fracture modes [1315], most of which, however, require comprehensive testing. In this study, a relatively simple plastic strain based criterion is used to model the housing material failure and good correlation is shown between FEA and testing. Element erosion [6] is employed to simulate the penetration of the housing due to the high speed impact from the wheel fragments, as shown in Figure 10. It is clear that the blades detach from the wheel at the region near the weakening slot, gradually impacting the housing. The wheel fragments reach speed zero at the end of the impact and the housing successfully contains the wheel fragments in this case study.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 10 Wheel velocity plots of the housing containment FEA model
6.
COMPARISON BETWEEN TEST AND SIMUALTION
The explicit FEA method of modelling the wheel burst and housing containment is applied to other case studies, where good correlation between test and simulation has been achieved. Figure 11 compares the wheel burst FEA with test by using the straight cut weakening slots. Again it demonstrates the importance of the weakening slot design. In the case study 3 shown in Figure 11(a), all three blades detach from the hub. Nevertheless, by using the slot design of case study 4, as shown in Figure 11(b), the middle blade remains intact, indicating that the slot depth may not be deep enough to result in the detachment of all three blades. Figure 12(a) and Figure 12(b) illustrate the correlation in terms of the damage at the inner wall and inlet of two turbine housings, respectively. Furthermore, the technique of turbine housing containment FEA has been applied to the compressor containment with Aluminum impeller and ductile iron housing. Figure 13 showcases the correlation of the fractured impeller from the compressor containment test and FEA simulation.
(a) Case study 3
(b) Case study 4 Figure 11 Comparison of turbine wheel burst test and FEA
(a) Case study 5
(b) Case study 6 Figure 12 Comparison of turbine housing containment test and FEA
Figure 13 Comparison of impeller burst in compressor containment test and FEA
7.
CONCLUSION
In this paper two major aspects of the turbine containment, i.e. wheel burst and housing containment, have been simulated by explicit FEA; which demonstrates excellent correlation with testing. Based on this study, the following conclusions may be drawn: a) Wheel burst tends to occur if increasing the outer diameter or depth of the weakening slot. b) In the process of optimising the weakening slot, it is essential to find a weakening slot design which can burst the wheel at exactly the target burst speed. It is believed that if the outer diameter or the depth of the slot is set to be too large, the wheel would burst at a speed lower than the target speed and vice versa. c) The Johnson Cook strength model is suitable to simulate the plastic behaviour of turbine housing, which is subjected to large strain, high strain rate and high temperature in the containment simulation. d) A plastic strain based failure criterion combined with element erosion technique may be capable of modelling the ductile material failures in turbo containment FEA.
ACKNOWLEDGEMENTS The authors would like to thank Cummins Turbo Technologies to allow publishing this paper. Our sincere thanks also go to Mr. Owen A Ryder, Mr. Jeffrey D Jones and Mr. Simon J Tooley who provided many valuable suggestions and proof read this paper. We would like to acknowledge the consistent support from other team members of the Applied Mechanics department. REFERENCE LIST 1)
2) 3) 4) 5) 6) 7)
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