Power Magnetic Devices: A Multi-Objective Design Approach Chapter 10: Introduction to Thermal Equivalent Circuits
10.1 Heat Thermal energy density and thermal energy Notation: spatial average
10.1 Heat Mean temperature and thermal capacitance of a region
10.1 Heat Heat flux and Fourier’s Law Note: comparison of heat flux and magnetic flux density
10.1 Heat Heat transfer rate Heat transfer
10.1 Heat The heat equation
10.1 Heat Derivation of heat equation
10.1 Heat Derivation of heat equation (continued 1/4)
10.1 Heat Derivation of heat equation (continued 2/4)
10.1 Heat Derivation of heat equation (continued 3/4)
10.1 Heat Derivation of heat equation (continued 4/4)
10.2 TEC of One-Dimensional Heat Flow Consider a region W For steady state conditions, we can show
10.2 TEC of One-Dimensional Heat Flow Derivation
10.2 TEC of One-Dimensional Heat Flow Derivation (continued)
10.2 TEC of One-Dimensional Heat Flow Mean temperature Derivation
10.2 TEC of One-Dimensional Heat Flow Derivation (continued)
10.2 TEC of One-Dimensional Heat Flow Boundary conditions Derivation
10.2 TEC of One-Dimensional Heat Flow Derivation (continued)
10.2 TEC of One-Dimensional Heat Flow Static TEC
10.2 TEC of One-Dimensional Heat Flow Derivation (1/3)
10.2 TEC of One-Dimensional Heat Flow Derivation (2/3)
10.2 TEC of One-Dimensional Heat Flow Derivation (3/3)
10.2 TEC of One-Dimensional Heat Flow Dynamic portion of TEC Derivation
10.2 TEC of One-Dimensional Heat Flow Derivation (continued)
10.2 TEC of One-Dimensional Heat Flow Example 10.2B. Consider a bar as follows: Dimensions: 10 cm (length) by 2 cm by 2 cm Specific heat capacity: 469 J/kg·K Mass density 7500 kg/m3 Thermal conductivity: 15.7 W/m·K Ends of bar at 26 oC Power dissipation in the bar is 10 W Compare transient response of TEC and direct solution of PDF
10.2 TEC of One-Dimensional Heat Flow Mean temperature versus time
10.2 TEC of One-Dimensional Heat Flow Heat transfer rate versus time
10.2 TEC of One-Dimensional Heat Flow Temperature profile versus time
10.2 TEC of One-Dimensional Heat Flow Peak temperature in region
10.2 TEC of One-Dimensional Heat Flow Derivation
10.2 TEC of One-Dimensional Heat Flow Example 10.2B. Let us find the peak temperature for steady state conditions for Example 10.2A. We obtain: TWcx: 63.4 oC < TW >: 38.5 oC TW,pk: 44.7 oC Important -
10.2 TEC of One-Dimensional Heat Flow TEC for one dimensional element w/o heat production or substantial mass
10.3 Thermal Equivalent Circuit of Cuboid Assumptions for cuboid region Assumed temperature distribution
10.3 Thermal Equivalent Circuit of Cuboid Mean temperatures
10.3 Thermal Equivalent Circuit of Cuboid Terms of interest
10.3 Thermal Equivalent Circuit of Cuboid Derivation
10.3 Thermal Equivalent Circuit of Cuboid Static TEC
10.3 Thermal Equivalent Circuit of Cuboid Derivation (1/3)
10.3 Thermal Equivalent Circuit of Cuboid Derivation (2/3)
10.3 Thermal Equivalent Circuit of Cuboid Derivation (3/3)
10.3 Thermal Equivalent Circuit of Cuboid Dynamic portion of TEC
10.3 Thermal Equivalent Circuit of Cuboid Derivation
10.3 Thermal Equivalent Circuit of Cuboid Peak temperature Derivation
10.3 Thermal Equivalent Circuit of Cuboid Derivation
10.4 Thermal Equivalent Circuit of Cyld. Region Consider a cylindrical region
10.4 Thermal Equivalent Circuit of Cyld. Region Conversion of heat equation to cylindrical coordinates
10.4 Thermal Equivalent Circuit of Cyld. Region Neglecting tangential heat flow Assumed temperature profile
10.4 Thermal Equivalent Circuit of Cyld. Region Radial portion of heat equation. We can show that Derivation
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (continued)
10.4 Thermal Equivalent Circuit of Cyld. Region Spatial average Mean temperature
10.4 Thermal Equivalent Circuit of Cyld. Region Axial heat flow
10.4 Thermal Equivalent Circuit of Cyld. Region Radial heat flow. First, we can show
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation
10.4 Thermal Equivalent Circuit of Cyld. Region We can also show
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation
10.4 Thermal Equivalent Circuit of Cyld. Region Finally we obtain
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (1/3)
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (2/3)
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (3/3)
10.4 Thermal Equivalent Circuit of Cyld. Region Dynamic portion of TEC of cylindrical region
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation
10.4 Thermal Equivalent Circuit of Cyld. Region Final TEC of cylindrical region
10.4 Thermal Equivalent Circuit of Cyld. Region Peak temperature
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (1/2)
10.4 Thermal Equivalent Circuit of Cyld. Region Derivation (2/2)
10.5 Inhomogeneous Regions It is often the case that we wish to develop a homogenized representation of a material
10.5 Inhomogeneous Regions Effective thermal properties in xy-direction Effective properties in z-direction Effective density and heat capacity
10.5 Inhomogeneous Regions Derivation in xy-direction (1/3)
10.5 Inhomogeneous Regions Derivation in xy-direction (2/3)
10.5 Inhomogeneous Regions Derivation in xy-direction (3/3)
10.5 Inhomogeneous Regions Derivation in z-direction (1/2)
10.5 Inhomogeneous Regions Derivation in z-direction (2/2)
10.5 Inhomogeneous Regions Derivation of effective density
10.5 Inhomogeneous Regions Derivation of effective heat capacity
10.5 Inhomogeneous Regions Example 10.5A. Let us compute the peak temperature in a slot of conductors. Slot is 2.4 cm wide, 2.8 cm deep, 7.1 cm long. Slot contains 50 conductors of 11 gauge copper wire with radius of 1.15 mm and insulation of 34.3 mm. Current density is 7.5 A/mm2. Thermal conductivity of copper, insulation, and air are 400 W/K·m, 0.175 W/K·m and 0.024 W/K·m. Walls of slot are 50 oC, ends of slot are 52 oC, and top of slot is 53 oC.
10.5 Inhomogeneous Regions Example 10.5A continued (1/2)
10.5 Inhomogeneous Regions Example 10.5A continued (2/2)
10.5 Inhomogeneous Regions Example 10.5A continued (2/2)
10.5 Inhomogeneous Regions Example 10.5A temperature profile
10.6 Material Boundaries Contact resistance For steel to steel: hcv ~ 1 kW/K·m2 at 5 MPa
10.6 Material Boundaries Convective heat transfer Natural convection in air: hcv ~ 2-10 W/k·m2 Natural convection in water: hcf: ~ 200 W/k·m2
10.6 Material Boundaries Radiation emission from object Radiation absorption by object Stefan-Boltzmann constant s: 56.7 nW/K4·m2 Emissivity es is between 0 and 1
10.6 Material Boundaries Net power due to radiation
10.7 Thermal Equivalent Circuit Networks Thermal equivalent circuit laws: The sum of the changes in temperature around a closed loop must be zero
10.7 Thermal Equivalent Circuit Networks Thermal equivalent circuit laws: the sum of the heat transfer rates into a node must be zero Mathematically constructed nodes Physical nodes
10.7 Thermal Equivalent Circuit Networks Standard branch
10.7 Thermal Equivalent Circuit Networks Origin of dependent source
10.7 Thermal Equivalent Circuit Networks Nodal analysis forms system of equations of form Reasons to use nodal analysis (instead of mesh)
10.7 Thermal Equivalent Circuit Networks Nodal analysis formulation algorithm
10.7 Thermal Equivalent Circuit Networks Derivation (1/3)
10.7 Thermal Equivalent Circuit Networks Derivation (2/3)
10.7 Thermal Equivalent Circuit Networks Derivation (3/3)
10.7 Thermal Equivalent Circuit Networks Graphical shorthand for describing elements
10.7 Thermal Equivalent Circuit Networks More graphical shorthand
10.8 Case Study: Electromagnet Cuboids of electromagnet
10.8 Case Study: Electromagnet Representing a rounded corner
10.8 Case Study: Electromagnet Winding to core resistance
10.8 Case Study: Electromagnet Derivation (1/2)
10.8 Case Study: Electromagnet Derivation (2/2)
10.8 Case Study: Electromagnet Airgap thermal resistance
10.8 Case Study: Electromagnet Thermal equivalent circuit
10.8 Case Study: Electromagnet Thermal resistances to ambient
10.8 Case Study: Electromagnet Electro-thermal analysis (1/2)
10.8 Case Study: Electromagnet Electro-thermal analysis (2/2)
10.8 Case Study: Electromagnet Solution algorithm
10.8 Case Study: Electromagnet Solution algorithm (continued)
10.8 Case Study: Electromagnet Example 10.8A. Revisit electromagnet design
10.8 Case Study: Electromagnet
10.8 Case Study: Electromagnet Perform thermal analysis for Design 250 of Section 5.4.
10.8 Case Study: Electromagnet Thermal data
10.8 Case Study: Electromagnet Convergence: e = 150 K, 37.1 K, 6.42 K, 1.16 K, 0.208 K, 37.5 mK, and 6.7 mK Maximum node temperature: 417 K or 144 oC Maximum device temp (element K): 379 K or 106oC Maximum surface temp: 338 K or 65 oC
10.8 Case Study: Electromagnet Impact of temperature rise Coil resistance increases from 4.82 W to 5.99 W Coil current drops from 2.49 A to 2.00 A
10.8 Case Study: Electromagnet Example 10.8B. Revise design process to include thermal analysis Steps: Remove current density constraint Add peak temperature constraint on winding
10.8 Case Study: Electromagnet Recall Constraints 1. Conductors fit 2. Packing factor 3. Current density 4. Aspect ratio 5. Volume 6. Loss 7. Force Metrics Volume and loss
10.8 Case Study: Electromagnet Pseudo-code
10.8 Case Study: Electromagnet Pseudo-code (continued)
10.8 Case Study: Electromagnet Pareto-optimal front
10.8 Case Study: Electromagnet Gene distribution
10.8 Case Study: Electromagnet Current density
10.8 Case Study: Electromagnet Conductor counts
10.8 Case Study: Electromagnet Widths
10.8 Case Study: Electromagnet Assorted dimensions
10.8 Case Study: Electromagnet Peak winding temperature
10.8 Case Study: Electromagnet Design 65 (with thermal analysis) Design 250 (without thermal analysis)
10.8 Case Study: Electromagnet Improvements