Thermal models of components for preliminary design of more electrical aircraft systems

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Thermal models of components for preliminary design of more electrical aircraft systems

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application/pdf Thermal models of components for preliminary design of more electrical aircraft systems lorian Sanchez, Marc Budinger, Ion Hazyuk
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Thermal models of components for preliminary design of more electrical aircraft systems

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Thermal models of components for preliminary design of more electrical aircraft systems. Florian Sanchez (1), Marc Budinger (2), Ion Hazyuk (3) 1 : Institut Clément Ader, 3 rue Caroline Aigle 31400 Toulouse, florian.sanchez@insa-toulouse.fr 2 : INSA/Institut Clément Ader, mbudinge@insa-toulouse.fr 3 : INSA/Institut Clément Ader, ion.hazyuk@insa-toulouse.fr Abstract The more electric aircraft shows a trend of the use of electrical power onboard of embedded systems. However, this technology also keeps its flaws - the high thermal constraints, which were not present in the hydraulic technology. This change requires the integration of these new constraints in the preliminary design of the system. Today, for the thermal simulation we have a wide range of tools, based on finite elements/volumes method. However, they require a defined geometry, which makes them unsuitable for the preliminary design and sizing. This paper presents a new form of metamodel called Scaling-Law-based-metamodels (SLAWMM), and its associated construction adapted to the preliminary design of more electrical aircraft systems (actuators, power electronics). The proposed meta-modeling method uses scaling laws to provide compact design models out of local numerical simulations. Compared to the traditional metamodels (polynomial response surfaces, kriging or radial basis function), the scaling-law-based metamodels have the advantage of a light, compact form and good predictive accuracy across a wide variation range of the design variables. Introduction The aim of the methodology described in this paper is to generate models adapted for the preliminary design while providing the accuracy close to 3D models [1]. This methodology for regression [2] is based on scaling laws [3] and dimensional analysis. Due to its physical foundation provides relatively simple mathematical forms while presenting extrapolation capabilities. After a short description of the methodology used for the scaling law generation, two examples are presented for typical components of an Electro- Mechanical Actuator (EMA) and power electronics. 1/ Scaling laws based meta-models 1.1 Model description Scaling laws are based on dimensional analysis [4], also called similarity laws or allometric models, and have been successfully used during the past decades for solving scientific and engineering problems [5]–[7] and for presenting results in a compact form. Generally the mathematical form of scaling laws is a power law, Budinger [2] proposed a modified form of scaling law, in order to use it for the design of mechatronics systems: , , … , ,… (1) where : • y is the parameter to be estimated; • L is the characteristic length of the system; • , , … and , , … are functions of the dimensionless numbers of the system. This mathematical form has the advantage of a simple and compact formulation, and it has physical foundations throughout its dimensionless numbers. The multiplicative and the power functions, k and a, are polynomials or power laws function of dimensionless numbers. The methodology proposed by Budinger [2] gives several criteria for the choice of a mathematical function to be used for each dimensionless number. 1.2 Model construction According to this approach, the metamodel is obtained in three main steps, described in Fig. 1: Fig. 1 : SLAWMM construction process • ‘’Data generation’’: First, the number of variables can be reduced via a sensitivity analysis. Then, a dimensional analysis is conducted to reduce the dimension of the problem. Finally, the design of experiments (DoE) [8] (full factorial design) is generated for the dimensionless numbers and implemented in a simulation software. • ‘’Metamodel form definition’’ : the form of the multiplicative and power functions is defined using the criteria defined by methodology [2]. • ‘’Metamodel building’’ : First a linear regression on the simulation data is performed to obtain the numerical values of the numerical coefficients of the multiplicative and power function. Then, the results of the linear regression are used as a start point for the non-linear regression. Finally, the quality of the estimation model can be evaluated using “X=Y diagram”, maximum relative error, robustness etc. 2/ Applications for electromechanical actuator and power electronics components 2.1 Mechanical housing Generally, a common housing integrates the components of an EMA and thus plays the role of the thermal interface with the surrounding environment (i.e. the air inside the plane wing) (Fig. 2). An estimation of the equivalent thermal resistance between the EMA housing and the air surrounding the plane would be very useful during preliminary design. Unfortunately, such models are not available. Therefore, the scaling-law-based-metamodels methodology can be applied to estimate this thermal resistance. Fig. 2 : Aileron actuator integration Although the EMA geometry may be complex, several simplifications can be made for preliminary design (Fig. 3). Usually, aileron EMA’s can be assimilated to an assembly of two joined cylinders. Since aircraft manufacturers prohibit, during the design, the heat exchange on the EMA extremities, the study can be conducted in 2D (the heat is exchanged only by its lateral surface). Another considered assumption is a perfect thermal bridge between the two cylinders. Thus the problem can be reduced to the study of one cylinder with a symmetry hypothesis. This cylinder is placed in a rectangular box in order to simulate the confined environment of the wing. Fig. 3 : Steps of geometrical simplifications Fig. 4 : Geometrical configuration and boundary conditions Figure 4 shows the geometrical and boundary condition parameters. The heat transfer coefficients and model the heat exchange with the air flow around the wing. The other parameters on which depends the problem are the properties of the air inside the wing. • ‘’Data generation’’: A sensitivity analysis on the identified parameters shows that the heat transfer coefficient has a small influence on the researched thermal resistance. The dimensional analysis shows that the problem depends on five dimensionless numbers: , π ! , π" # , $ $ % , & '()*$. Considering the cylinder diameter as the characteristic length, the mathematical expression expected for the thermal resistance is: )*$ , !, #, $ ( +,, -, ., / (2) The used DoE has four levels for the dimensionless numbers and three levels for the cylinder diameter. The finite element simulations were performed in COMSOL software (Fig. 5). Fig. 5 : Housing temperature field [in °K] • ‘’Metamodel form definition’’: The simulation data analysis according to the methodology led to the expressions (3) and (4) for the power function , !, #, $ and the multiplicative coefficient , !, #, $ respectively. !, #, $ 0 ! 1 0 ! 1 2 # 1 2 # 1 3 $ 1 ( (3) , !, #, $ 45 1 4 ! 1 4 ! 1 46 # 1 47 # 1 48 ! # ⋅ : $ ;< = ⋅ ;> (4) where: 0?, 2?, 3, ( @A 4? are numerical coefficients. • ‘’Metamodel building’’: the numerical coefficients are calculated by (first) linear and (then) non-linear regressions. The obtained model is validated by using an X=Y diagram (see Fig. 6). Fig. 6 : X=Y diagram of the equivalent thermal resistance of the EMA As it can be seen, the model has a maximum error of 19% and the mean error is about 5%, which are acceptable in early design stage. 2.2 Capacitor Another example of SLAWMM application is the evaluation of the equivalent thermal resistance of a capacitor (Fig. 7 : Film capacitor (AVX)Fig. 7). Indeed, there is a real need for thermal model in the field of power electronics [9], [10]. The particularity of this example is that the heat is dissipated by all the three heat transfer modes (conduction, convection and radiation). Indeed, for this geometry estimation laws are available for each heat transfer mode. But, in order to use them we need to estimate the hot-spot temperature and the surface temperature. And to solve it one needs to use iterative algorithms. The preliminary designer, however, is interested in a straight forward relation that estimates the equivalent thermal resistance between the hot-spot and the ambient temperature. Fig. 7 : Film capacitor (AVX) For the detailed modeling of the capacitor, two simplifying assumptions are considered (Fig. 8): 1.Geometrical simplification: axi-symmetry hypothesis. 2.Material simplification: An equivalent homogenized material is considered instead of modeling all the dielectric and electrode layers. The equivalent material properties are thus functions of the layers physical and geometrical characteristics. Fig. 8 : Simplifying assumption for capacitor detailed modeling • ‘’Data generation’’: The dimensional analysis shows that the problem depends on five dimensionless numbers: ∗ ∗ ∗ ∗ , # , %CDED % , 6 λDRIJ, 7 % KLMN . The last dimensionless number 7 is a function of the ambient temperature. In order to reduce the number of variable, the study is made for a fixed ambient temperature (O?PQ 25°3 ; so this dimensionless number is not considered. Considering the cylinder diameter as the characteristic length, the mathematical expression expected for the thermal resistance is: )*$ , , ( +,, , (5) The used DoE has three levels for the dimensionless number and for the capacitor diameter. The finite elements simulations are performed on COMSOL software (Fig. 9). Fig. 9 : Capacitor temperature field [in °K] • ‘’Metamodel form definition’’: The simulation data analysis according to the methodology led to the expressions (6) and (7) for the power function , , and the multiplicative function , , respectively. , , 5 1 1 1 6 1 7 (6) , , 5 1 1 6 ⋅ : UV = ⋅ UW (7) Where: ? and ? are numerical coefficients. • ‘’Metamodel building’’: The numerical coefficients are calculated by (first) linear and (then) non-linear regressions. The obtained model is validated by using an X=Y diagram (see Fig. 10 : X-Y diagram of the thermal resistance of the capacitor Fig. 10 : X-Y diagram of the thermal resistance of the capacitor As it shown, the model has a maximum error of 20% and the mean error is about 5% which are acceptable in early design stage. Conclusions The more electric aircraft requires new estimation models to consider the thermal constraints during the sizing of the new generations of systems. The article has shown that the SLAWMM methodology mays offer the possibility to build thermal model of complex configurations. Furthermore, the methodology is based on the dimensional analysis what gives a physical insight for the obtained models. References [1] M. Budinger, A. Reysset, T. El Halabi, C. Vasiliu, J. Maré, U. De Toulouse, I. Ups, and I. C. Ader, “Optimal preliminary design of electromechanical actuators,” vol. 33, no. 0, pp. 1–22. [2] M. Budinger, J.-C. Passieux, C. Gogu, and A. Fraj, “Scaling-law-based metamodels for the sizing of mechatronic systems,” Mechatronics, Dec. 2013. [3] M. Budinger, J. Liscouët, F. Hospital, J. Maré, U. De Toulouse, I. Ups, and I. C. Ader, “Estimation Models for the Preliminary Design of Electro-Mechanical Actuators,” vol. 3, pp. 243–259, 2012. [4] E. S. Taylor, Dimensional Analysis for Engineers. Oxford University Press, 1974. [5] P. F. Mendez and F. Ordóñez, “Scaling Laws From Statistical Data and Dimensional Analysis,” J. Appl. Mech., vol. 72, no. 5, p. 648, 2005. [6] G. Miragliotta, “The power of dimensional analysis in production systems design,” Int. J. Prod. Econ., vol. 131, no. 1, pp. 175–182, May 2011. [7] J. I. Prieto and a B. Stefanovskiy, “Dimensional analysis of leakage and mechanical power losses of kinematic Stirling engines,” Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., vol. 217, no. 8, pp. 917–934, Jan. 2003. [8] J. Goupy, “Plans d ’ expériences,” Tech. l’Ingénieur, vol. PE 230, 1997. [9] M. Andresen and M. Liserre, “Impact of active thermal management on power electronics design,” Microelectron. Reliab., pp. 5–9, Aug. 2014. [10] P. Cova and N. Delmonte, “Thermal modeling and design of power converters with tight thermal constraints,” Microelectron. Reliab., vol. 52, no. 9–10, pp. 2391–2396, Sep. 2012. 0 5 10 15 20 25 0 5 10 15 20 25 Rth_SLAWMM [K/W] Rth_COMSOL [K/W] Maximum error : 19.75% Average error : 4.73%