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CS Thermal Network Method in the Design of Electric Power Equipment Christoph Gramsch1, Andreas Blaszczyk2, Helmut Löbl1 and Steffen Grossmann1 1 Technical University Dresden, Institute of Electrical Power Systems and High Voltage Engineering, 01062 Dresden, Momsenstr. 10, Germany 2 ABB Corporate Research, 5405 Baden-Daettwil, Switzerland 1 {CGramsch, Loebl, Grossmann}@ieeh.et.tu-dresden.de , [email protected] Abstract— The paper presents basic principles of the thermal networks method. Modeling of network elements has been shown for convection and radiation. A concept of hierarchical thermal network models for complex geometries has been explained. An example of the thermal network computation including comparison with test results has been presented. Keywords— thermal networks, thermal design, power devices I. INTRODUCTION The Thermal Network Method (TNM) is based on a substitution of an arbitrary 3D geometry by a circuit consisting of thermal resistances, capacitances and heat sources. For such a network the currents correspond to heat flow and the nodal potentials to temperatures. Due to similarity of mathematical formulations the electrical circuit programs can be used to obtain a solution. The basic advantage of the thermal network analysis is the fast computation time: steady state computations of large models can be performed within a few seconds. Therefore the TNM is very suitable for parameter studies and become popular as a tool supporting the industrial design. A drawback of TNM is the creation of the network, in particular transition from the real geometry to a network based model. This drawback can be mitigated by applying hierarchical modelling approach and reusable library elements with readyto-use representations of the whole devices. In this paper we describe such an approach and its application to real cases. II. numbers of Nusselt (Nu), Grasshof (Gr), Prandtl (Pr) and Reynolds (Re). The following basic relationship can be used conv Nu fluid l ch (2) where fluid is the thermal conductivity of fluid and lch the characteristic length (e.g. the height of a vertical plate or the diameter of a horizontal cylinder). The Nusselt number is calculated for the natural convection with coefficients c1 and n1: Nu c1 Gr Pr 1 n (3) while for the forced convection with coefficients c2 and n2: Nu c2 Ren2 . (4) The implemented convection resistances can be used for the laminar and turbulent flow models in different fluids (air, SF6, oil, H2O). generated heat P=I2R LX L conduction through insulation conduction along conductor BASIC CONCEPT The basic concept of substituting the geometrical objects by a thermal network model is shown for an example of a coated conductor carrying electrical current I, Fig. 1. The current generates temperature dependent power losses that are conducted along the conductor (in copper) as well as through the insulation layer and dissipated via convection and radiation. A part of the generated heat can be stored in the conductor material represented by the capacitance C (used for the transient computations only). In this abstract we present basic formulas for the calculation of the circuit elements corresponding to convection and radiation. The thermal resistance of convection can be expressed as follows R conv 1 conv A conv (1) with the convection coefficient conv and the surface area Aconv. The calculation of conv is based on the similarity theory [1,2,3], which requires evaluation of the characteristic radiation convection R S C Fig. 1. Thermal network model of a coated conductor The radiation resistance is expressed in a similar way as (1): R rad 1 . rad A rad (5) The radiation coefficient rad is based on the StefanBoltzmann-constant , the emissivity 12 between the radiating and absorbing surfaces and the absolute temperatures T1 and T2 of both surfaces: rad 12 T14 T24 T1 T2 (6) CS The value of 12 depends on emissivity of the emitting and absorbing surfaces, their surface area and the viewing factor between them. In case of a conductor located in a free space the emissivity of the conductor outer surface can be used for 12. created model allows simulation of mass transfer phenomena and has been used to study different ventilation solutions. a) 8 7 Due to the temperature dependency of convection and radiation resistances as well as power losses the network problem is nonlinear. This kind of problems can be efficiently solved using programs for analysis of electric circuits like Spice or its commercial derivates (www.pspice.com). 6 5 HIERARCHICAL APPROACH The generation of a thermal network for a complex device is a time consuming process. To make it easier and faster we introduced a concept of hierarchical thermal networks. In hierarchical approach we create networks consisting not only of primitive elements representing physical phenomena like resistors or sources but also models of whole components. For example, the network scheme of a conductor shown in Fig. 1 consists of 5 thermal resistances, one capacitance and one source. These elements can be wrapped together into a new element representing the whole conductor with pins corresponding to the heat conduction (L,LX), convection (C), radiation (R) and the outer surface (S). The example in Fig. 2 shows application of the new “coated conductor” element (denoted here as CN_gICYL1) in a network model representing encapsulated conductor. In this model additionally radiation, convection and eddy losses related to enclosure walls as well as ventilation have been included. The network from Fig. 2 can be again wrapped into a new “encapsulated conductor” element and applied in higher hierarchical levels. For complex devices we use up to 5 hierarchical network levels. The hierarchical approach allows a better management of large models and reusability of components. LX L 3 2 1 b) 90 Temperature [°C] III. 4 left phase 75 mid right phase phase Dotted lines = measured temperatures 60 3 1 R C Fig. 2. Hierarchical thermal network with a coated conductor element IV. 8 CONCLUSION Thermal network method has been effectively applied to thermal design of complex power devices like high and medium voltage switchgear, transformers, bushings and circuit-breakers. The hierarchical modeling approach allows efficient handling of complex geometries. A good agreement with experimental results can be achieved. The fast computation times enable comprehensive parameter studies for modeled devices. EXAMPLE An example of a complex power device is shown in Fig. 3a. Based on hierarchical thermal network approach we computed steady state temperatures along the phase conductors. The deviations between computations and tests are for most measured points in the range of 3 K, see Fig. 3b. The 6 Fig. 3. Air insulated medium voltage switchgear arrangement: a) geometry view b) comparison between measured and computed steady-state temperatures V. S Position VI. [1] [2] [3] REFERENCES J. P. Holman, “Heat Transfer,” McGraw-Hill Higher Education, 9th edition, 2002 H. Böhme, “Mittelspannungstechnik,” Verlag Technik Berlin München, 2. Auflage, 2005 H. Löbl, “Basis of Thermal Networks,” (unpublished) Dresden University of Technology, 1999.