Reliability Prediction for a Thermal System Using CFD and FEM Simulations

Transcription

1 Proceedings of IMECE ASME International Mechanical Engineering Congress and Exposition October 31 - November 6, 2008, Boston, Massachusetts, USA IMECE Reliability Prediction for a Thermal System Using CFD and FEM Simulations O.M. Al-Habahbeh 1, D.K. Aidun 2, P. Marzocca 3, and H. Lee 4 1, 2, & 3 Mechanical & Aeronautical Eng. Dept., Clarkson University, PO Box 5725, Potsdam, NY USA 1 Corresponding author, alhabaho@clarkson.edu, 2 daidun@clarkson.edu, 3 pmarzocc@clarkson.edu 4 GE Energy, Aero Package Engineering, Houston, TX USA hauhua.lee@ge.com Keywords: Reliability Analysis, Fatigue Life, Monte Carlo Simulation, CFD Simulation, FEM Simulation, Physics-based Modeling ABSTRACT A general procedure for reliability prediction is introduced. The procedure is applied to a cylindrical ring and can be used for any similar thermal application. The procedure is classified as a physics-based reliability prediction method. It utilizes different computational tools such as Computational Fluid Dynamics (CFD), Finite Element Method (FEM), and Monte Carlo Simulations (MCS). The process starts with CFD simulation to find the convective terms necessary for the transient FEM thermal analysis. The transient FEM thermal analysis provides values for thermal stress. These values are used in the fatigue life analysis. The end result is the fatigue life of the component. As a result of input parameters uncertainty, the resulting life will be in the form of a Probability Density Function (PDF), which enables the calculation of the reliability of the component. ACRONYM FEM: Finite Element Method FSI: Fluid Solid Interaction CFD: Computational Fluid Dynamics MCS: Monte Carlo Simulation CEL: CFX Expression Language CCL: CFX Command Language PDF: Probability Density Function CDF: Cumulative Distribution Function LHS: Latin Hypercube Sampling AHTC: Air Heat Transfer Coefficient WHTC: Water Heat Transfer Coefficient ID: Inside Diameter OD: Outside Diameter NOMENCLATURE Convection Ambient Temperature: Air Temperature Convection 2 Ambient Temperature: Water Temperature σ 1 : First Principal Stress (MPa) σ 2 : Second Principal Stress (MPa) σ 3 : Third Principal Stress (MPa) σ y : Yield Stress (MPa) σ Thermal : Thermal Stress (MPa) E: Modulus of Elasticity α: Coefficient of thermal expansion ΔT: Temperature gradient c: Constant of proportionality INTRODUCTION Most deterministic design methods are combined with safety factors that are assigned based on experience, and consequently the resulting design is not optimized and its reliability is not known. On the other hand, unlike traditional point design methods, reliability engineering provides

2 optimum and reliable designs. Reliability is defined as the ability of a system to operate under normal and by opposite, abnormal conditions subject to a defined failure ratio and for a specific life time [1]. Reliability evaluation of thermal systems is of a paramount importance for predicting their performance. It provides a wealth of information that can be used to improve the quality of their service. The availability of reliability data helps to plan more effective maintenance and warranty. The reliability engineering method entails a robust design process which contains a point design sub-process conducted using CFD and FEM. In this work, reliability methods are employed to investigate the reliability of a thermal model. The procedure used for this model can be applied for more complicated thermal models. The present model is a cylindrical ring. Its reliability is evaluated using a physicsbased model, based on transient analysis. The computational tools used in this process are CFD, FEM, and MCS. The process includes building a CFD model, obtaining heat transfer film coefficients, using the CFD results for FEM analysis, and obtaining structural thermal stresses. These stresses cause cyclic thermal fatigue; therefore, they are identified as the cause of failure that is used to determine the reliability of the model. The process of reliability evaluation entails the integration of CFD and FEM tools with probabilistic tools in order to setup the reliability prediction loop. It can also be supplemented by considering application-specific deteriorateing factors such as corrosion rate. The degradation caused by these factors can be integrated with this procedure to evaluate the overall reliability. The reliability prediction loop used in this work is shown in Fig. 1. The procedure starts by conducting CFD analysis to obtain heat convection coefficients. MCS iterates the CFD loop to obtain mean values of the heat transfer coefficients. The mean values are used as part of the input to the FEM analysis. MCS generates inputs for the FEM analysis, which is iterated resulting in thermal stress and fatigue life. These results form response surfaces that are used to generate large numbers of points. Life distribution and tabular data are obtained using these points. Reliability can be determined using the resulting tabular data. Both environmental and operational envelopes are utilized in the simulation process, where uncertainties in environment, operation, and manufacturing, such as variation in temperatures and dimensions are considered. PHYSICS-BASED MODELING Physics-based modeling is used in this work in conjunction with reliability methods. The main interest in this research is to analyze the transient start up of the model, and to achieve this goal, the system is modeled using transient analysis. The thermal stresses developed in the model are the result of temperature gradients resulting from heat flux. The model of interest is a cylindrical ring. It is modeled and simulated using CFD and FEM. The fluids involved are 2 water and air. The ring is built as shown in Fig. 2. CFD analysis is conducted on the model, and convection coefficients due to heat transfer across the ring are obtained. The FEM simulation includes utilizing the CFD results for the FEM model and obtaining structural thermal stresses. These stresses are used to conduct fatigue analysis in order to predict life of the model. This work determines the convection heat transfer coefficients for water inside the ring and for air outside the ring. While the mode of heat transfer within the ring is conduction. This method can be used to study similar problems in thermal components such as ducts and heat exchanger shells. The accuracy of the solution is verified by refining the mesh. Thermal stress in the model is determined using FEM. START Heat Convection Coefficients CFD ANSYS/CFX MCS/ Input Generation MCS/ Input Generation FEM ANSYS/ Simulation Thermal Stress & Fatigue Life MCS ANSYS/ DesignXplorer MCS ANSYS/ DesignXplorer RELIABILITY FIG. 1: RELIABILITY PREDICTION LOOP CFD SIMULATION Heat Convection Coefficients Mean Values Life Distribution ANSYS/CFX is used for the CFD simulation. The package has the capability of solving steady-state and transient problems in thermo-fluids [2]. It is used to simulate the fluid flow and the Fluid and Solid Interaction (FSI). The basic operation of CFX relies on utilizing control volume method, and implicit, pressure-based algorithms for all incompressible flow speed. The general post-processing capabilities include

3 batch processing, quantitative post-processing, in addition to length, area, volume and mass flow-based averaging and integration, and support of expressions [3]. ANSYS Simulation is the analysis tool to simulate the thermal stress and fatigue. Fig. 2 shows the ring model under consideration, with water as the inner fluid, and air as the outer fluid. Table 1 shows the dimensions of the model. The analyses include CFD, one-way thermal FSI, and stress solutions. The steps for conducting these analyses include building the geometry using ANSYS Workbench Design Modeler, meshing the model using CFX -Mesh, setting up the model and applying boundary conditions using CFX -Pre, and solving the steady-state and transient cases using CFX -Solver. Post processing of the results is done in CFX -Post. The CFD simulation uses conjugate heat transfer to model heat conduction across the ring wall. The domains of the model are: 1) Water running in the ring 2) Ring material (steel) 3) Ambient air. Air is assumed an ideal gas, and air pressure used for the CFD simulations is kpa. Water pressure is 700 kpa, while air and water inlet temperatures and water flow rate are variable parameters. The boundary conditions imposed on the CFD model are shown in Fig. 2 and listed below: Still Air TABLE 2: CFD MESH SENSITIVITY Mesh Size Error in AHTC (%) Error in WHTC (%) 4X X X Water-out Water-in 1- Air enclosure. 2- Water enters from the right side of the ring and leaves from the left side. 3- Ring inner wall, which is the interface between the ring and the water, with no slip boundary condition. 4- Ring outer wall, which is the interface between ring and air, with no slip boundary condition. 5- Temperature of air enclosure outer wall is equal to ambient air temperature. TABLE 1: MODEL GEOMETRIC DATA Dimension Value Unit Outer diameter 60 cm Inner diameter 50 Length 20 FIG. 2: CYLINRICAL RING MODEL Air Ring Water The Tetrahedron finite volume element is used for meshing the model, in addition to inflation layers for boundary layer refinement, and face proximity to achieve a minimum number of elements across the ring thickness. Fig. 3 shows part of the finite volume mesh. The mesh contains around 326,000 tetrahedral and prism elements. Before conducting any MCS s, the grid is verified by refining the mesh size three times. Table 2 shows the mesh sensitivity in calculating air and water heat transfer coefficients. FIG. 3: FINITE VOLUME MESH From Table 2, it appears that a reasonable mesh independency is achieved using 2x2 grid. The latter grid size is used for all subsequent CFD analyses. 3

4 MCS FOR CFD ANALYSIS Monte Carlo Simulation technique is used for reliability estimation of the thermal model; it comprises the following steps [8]: a- Defining the problem in terms of random variables. b- Quantifying the probabilistic characteristics of the random variables in terms of their PDFs. c- Generating the values of these random variables. d- Numerical experimentation for each set of realizations of all the random variables. e- Extracting probabilistic information from N such realizations. f- Determining the accuracy and efficiency of the simulation The reliability of the thermal model is based on fatigue life, which in turn is based on the thermal stress developed during the transient operation. Some of the potential factors that may control the stress distribution are: FIG. 4: PDF FOR ID OF THE MODEL 1- Air flow rate and temperature 2- Water flow rate and temperature 3- Dimensional tolerances of the ring Table 3 summarizes the statistical characteristics of the CFD input parameters. A sample of 10,000 points is generated from this data. For this purpose, LHS was used combined with an approximation technique. Figures 4 to 8 show the PDFs for selected input parameters that are used as variables in the CFD and FEM analysis. For all the CFD analyses, the air natural convection flow is assumed laminar, while k-ε turbulence model is assumed for water flow. FIG. 5: PDF FOR OD OF THE MODEL TABLE 3: INPUT PARAMETERS USED IN MCS/CFD Parameter Air Temp. Water Flow Water Temp. ( C) (kg/s) ( C) Weibull Exponent Weibull Characteristic Value Mean Standard Deviation Minimum Maximum Skewness E-2 Kurtosis Shannon Entropy FIG. 6: PDF FOR WATER FLOW 4

5 PARAMETERS AND EXPRESSIONS IN ANSYS/CFX Performing MCS on the CFD analysis is made possible by using different software tools. For example, CEL is used to define quantities such as average heat transfer coefficients of areas [9], while CCL is used to define parameters in CFX which are used in MCS [10]. As ANSYS/CFX is one of the modules of ANSYS/Workbench, it is necessary to define all variable parameters in ANSYS/Workbench. The parameter types used here include input, response, and derived parameters [3]. MCS/CFD RESULTS FIG. 7: PDF FOR AIR TEMPERATURE The results of the MCS/CFD analyses are summarized in Table 4. In addition, heat transfer coefficients for water and air are shown in Fig. 9 and 10 respectively. The PDF for the AHTC resulting from MCS is shown in Fig. 11, while the PDF for the WHTC is shown in Fig. 12. TABLE 4: MCS/CFD OUTPUT PARAMETERS Parameter Air HTC Water HTC (W/m 2 C) (W/m 2 C) Mean Standard Deviation 9.70E Minimum 6.00E Maximum 1.40E Skewness kurtosis Shannon Entropy FIG. 8: PDF FOR WATER TEMPERATURE The above input parameters are used in the reliability analysis as random variables. Their PDFs are used to generate input values for the MCS of the CFD. The number of random points generated is directly proportional to the number of input variables. In this work, 25 points are generated. Each of these points is used to run the CFD analysis. and the result is 25 sets of CFD random results. These random results forms an approximated response surface. The response surface is used to generate a sample output size of 10,000 points. Finally, the PDFs of the heat transfer coefficients are obtained from these results using MCS. Once this data is available, it is used as input to the FEM transient simulation, in addition to other input parameters as will be shown in the FEM analysis section. FIG. 9: WATER HEAT TRANSFER COEFFICIENT 5

6 The results of the transient thermal analysis are used for the structural analysis, which is conducted to calculate the maximum thermal stress. The calculated stresses include von Mises and maximum shear stress. von Mises failure criterion is defined as [4]: ( s s ) 2 ( s s ) 2 ( s s ) s 1, s 2 s 3 2 2s y (1) where, and are the principal stresses, and σ y is the yield stress for the ductile material. FIG. 10: AIR HEAT TRANSFER COEFFICIENT The thermal stress σ Thermal is generally expressed in the form [5]: σ Thermal = ce.. α. Δ T (2) where E = Modulus of Elasticity α = Coefficient of thermal expansion Δ T = Temperature gradient c = Constant of proportionality The constant of proportionality c depends on the condition of mechanical constraint, temperature distribution, and Poisson s ratio. The value of the maximum stress is used to predict model life based on fatigue analysis. Thermal stress has more effect on service life than mechanical stress. The effect can be as much as 2.5 percent lower cycles [6]. This factor is integrated into the results of this analysis. The mesh used for the FEM analysis is shown in Fig. 13. It consists of more than 9000 hexahedral elements. The grid was refined three times in order to reach acceptable mesh insensitivity. FIG. 11: PDF FOR AIR HEAT TRANSFER COEFFICIENT FIG. 12: PDF FOR WATER HEAT TRANSFER COEFFICIENT FEM MODELING The CFD results are imposed on the corresponding FEM model in ANSYS Simulation, a transient FEM simulation is set-up to determine the maximum thermal stress, and a fatigue analysis is conducted to estimate the life of the model. The steady state is reached approximately after 10 minutes (600 s). 6 FIG. 13: FEM HEXAHEDRAL MESH As shown in Table 5, a very low amount of error is reached for a test load, which indicates that the results are reasonably mesh- independent.

7 TABLE 5: MESH SESITIVITY Mesh Max Stress (MPa) Error coarse medium Fine Once a good mesh is obt ained, FEM simulation is started by defining a transient thermal analysis, which is used to conduct the structural analysis. It is not known in advance when the maximum stress occurs, therefore it is necessary to investigate the whole transient period and find the value of the maximum stress. For this purpose, the FEM thermal analysis is conducted for a time period of 1000 s. This number is arbitrary and used just to see when the steady state is reached. Fig. 14 shows the ring temperature variation with time. There are two lines representing the maximum value and the minimum value. The time span is 1000 s. It is noticed that the steady state is reached approximately at t = 625 s. The maximum stress is calculated for different points (including the steady state area). The maximum thermal stress is plotted against time in Fig. 15. It is noticed that the peak stress occurs close to zero, and consequently, the time values close to zero should be investigated. Therefore, a new transient analysis is conducted with a reduced time step to investigate the time below 20 seconds. The corresponding temperature variation is shown in Fig. 16, and the stress variation is shown in Fig. 17, where the maximum thermal stress occurs between 5 and 15 s. Upon investigation, it is found that the maximum stress occurs at t = 8.75 s. The temperature distribution at maximum stress is shown in Fig. 18 and the maximum thermal stress distribution at t = 8.75 s is shown in Fig. 19. Stress (MPa) Time (s) FIG. 15: MAX TRANSIENT THERMAL STRESS FIG. 16: MIN & MAX TRANSIENT TEMPERATURES 140 Stress (MPa) Time (s) FIG. 17: MAX TRANSIENT THERMAL STRESS FIG. 14: MIN & MAX TRANSIENT TEMPERATURES 7

8 FIG. 18: TEMPERATURE DISTRIBUTION AT MAX TRANSIENT THERMAL STRESS FIG. 20: FATIGUE LIFE DISTRIBUTION MCS FOR FEM ANALYSIS The factors affecting MCS include sampling technique, number of simulations, distributions characteristics, and responses configuration. LHS is used for this work, where fewer simulations are required to estimate the statistical characteristics of the model behavior. The parameters to be used as random variables are selected as well as their respective distributions. These parameters are shown in Table 6. The mean values of water and air heat transfer coefficients are calculated using the preceding MCS/CFD analysis and used in the FEM analysis. INPUT PARAMETERS VARIATION FIG. 19: MAX THERMAL STRESS DISTRIBUTION Based on the maximum thermal stress, fatigue life is calculated using the material S-N curve and the fatigue module provided in ANSYS Simulation. The type of loading used is zero based, that means the material expands due to temperature increase, and when the temperature goes back to normal, the material will go back to its original shape. In other words, there is no negative stress value as can be seen in Fig. 19. Fatigue life distribution is shown in Fig. 20. It is noted in Fig. 20 that life is minimum at the center of the ring, and that is due to the ends effects. Therefore, the result away from the edges is considered. Table 6 shows the selected FEM input parameters statistical characteristics. They include selected environmental and geometrical variables, such as fluid temperature and ring diameter. The choice of the PDF type and characteristics depends on the behavior of the input parameter. One of the most widely used distributions for this purpose is Weibull distribution. Figures 4, 5, 7, and 8 show the MCS/FEM input variables distributions, as they were used previously for the MCS/CFD analysis. They are all of Weibull distribution type. After the FEM input parameters are simulated as shown in Fig. 1, 25 Design points are generated from their respective distributions using LHS technique. These design points are used to construct corresponding response approximation surfaces. These surfaces are used to generate 10,000 sample points in order to calculate the FEM output parameters. 8

9 TABLE 6: FEM INPUT PARAMETERS CHARACTERISTICS Air Temp. ( C) Water Temp. ( C) Parameter ID OD (cm) (cm) Minimum Maximum Weibull Exponent Weibull Characteristic Value Mean Standard Deviation Skewness E Kurtosis Shannon Entropy preset, and the reliability is sought, the result can be found using Table 7. On the other hand, using the same table, life for a given reliability can be determined. MCS/FEM RESULTS One of the most important results of the MCS/FEM analysis is the histogram of the fatigue life, which is shown in Fig. 21, and the CDF of the same parameter shown in Fig. 22. FIG. 23: RESPONSE APPROXIMATION SURFACE TABLE 7: LIFE PERCENTILE Life (Years) Reliability (%) The data in Table 7 is plotted in Fig. 24 to represent Life PDF. FIG. 21: FATIGUE LIFE HISTOGRAM FIG. 22: FATIGUE LIFE CDF Fig. 23 shows a response approximation surface for life against two variables; air temperature and water temperature. The effect of these variables on life can be investigated using this figure. In this research, the main task is to find the reliability of the model with respect to a certain number of cycles. If the model life in terms of the number of cycles is 9 FIG. 24: FATIGUE LIFE PDF As an example for using the data in Table 7, for a life of 4 years, the reliability of the ring is 84%. COMPARISON WITH THE TRADITIONAL METHOD Reliability analysis leads to safer and more economic systems than the ones designed using traditional point design methods

10 [7], because it provides data that is not available for other traditional methods. Such data includes the relationship between life and reliability. While the proposed reliability method provides this information, the traditional point design method combined with safety factor does not. As a comparison, safety factor result is obtained for the same model and same date used for the reliability analysis. This result is shown in Fig. 25. The minimum safety factor is 1.8. This information is useful in the sense that it quantifies the safety of the structure. However, it does not reveal how long the structure will be safe, and what the probability of failure is. Therefore, the reliability analysis provides more information than the traditional design method. This information include the change of expected performance with time, which helps for maintenance and warranty planning. CONCLUSIONS The proposed reliability analysis gives insight into the service life of thermal components. It can be done at an early stage during design phase which helps in the product planning. The Monte Carlo Simulation is a powerful technique used to determine the reliability of simulated models. Due to the complexity of CFD and FEM computations, in addition to the large number of required sample points, MCS method is computationally very expensive. Therefore, LHS and response surfaces approximations should be used for accurate modeling. The transient heat flux through the ring induces a variable thermal stress. Therefore, the whole transient phase should be investigated in order to find the maximum stress. Reliability prediction using MCS provides more useful information about the expected performance of a product than traditional design methods. ACKNOWLEDGMENT The authors would like to acknowledge support for this research provided by GE Energy, Houston, TX. FIG. 25: SAFETY FACTOR RESULT Table 8 shows the statistical characteristics for some output parameters. These parameters are the results of MCS/FEM. Parameter TABLE 8: OUTPUT PARAMETERS Max. Temp. ( C) Equivalent Stress Max. (MPa) Life in Years Min. Safety Factor Min. Max. Shear Stress (MPa) Min Max Mean Standard Deviation Skewness -8.65E Kurtosis REFERENCES [1] B.A. Cullimore, Reliability Engineering and Robust Design: New Methods for Thermal/Fluid Engineering, C&R White paper, Littleton Colorado, USA [2] ANSYS, Inc. web site: [3] Release 11.0 Documentation for ANSYS Workbench [4] Hosford W. F., Mechanical Behavior of Materials, ISBN , Cambridge Press, 2005, pp83. [5] NEA Nuclear Science Committee, Nuclear Energy Agency Utilization and Reliability of High Power Proton Accelerators, Workshop proceedings, Mito, Japan, October 1998, ISBN: , pp 320. [6] Oberg, E., McCauley, C.J., Machinery's Handbook: A Reference Book for the Mechanical Engineer, Industrial Press Inc., ISBN , pp [7] GC Avontuur, Reliability Analysis in Mechanical Engineering Design, December 2000, Delft University Press, ISBN: , pp [8] A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley & Sons, Inc., [9] ANSYS Technical Briefs: ANSYS/CFX Flexible CFD Solutions [10] ANSYS CFX-Post User s Guide. Shannon Entropy