1. A NEW ORIGINAL UNCODITIONALY STABLE MIXED FINITE ELEMENT APPROACH IN TRANSIENT HEAT ANALYSIS WITHOUT DIMENSIONAL REDUCTION Dubravka Mijuca, Bojan Medjo Faculty of Mathematics, Department of Mechanics University of Belgrade dmijuca@matf.bg.ac.yu Seminar for Rheology, 15 Mart, 2005

2. Reference The Finite Element Method - Volume 1: The Basis; O.C. Zienkiewicz, R.L. Taylor Finite Element Procedures; K. J. Bathe On hexahedral finite element HC8/27 in elasticity, Mijuca D. Mijuca D, Žiberna A, Medjo B (2005) A new multifield finite element method in steady state heat analysis, Thermal Science, in press Cannarozzi AA, Ubertini F (2001) A mixed variational method for linear coupled thermoelastic analysis. International Journal of Solids and Structures. 38: 717-739 LUSAS Theory Manual 1, Version 13 STRAUS 7 Verification Manual ANSYS Verification Manual

3. 1 st Law of Thermodynamics Initial condition: Boundary conditions:

4. Heat Transfer Modes Conduction Convection Radiation

5. Conduction Fourrier’s Law (1822.) k - Thermal Conductivity

6. Thermal Conductivities Wood 0.05 Water 0.7 Glass 0.8 Steel10-20 Iron 80 Copper 400 Silver 450 k [W/mK] (Room Temperature)

7. Convection Convection involves the exchange of Heat between a Fluid and a Surface Natural Convection Forced Convection 1701 – Newton’s “Cooling Law” T, T 0 – Temperatures of the surface and the Fluid h C – Convective (Film) Coefficient

8. Convective Coefficient depends on: Temperature Difference; Fluid; Fluid Speed; Geometry of the Surface; Roughness of the Surface.

9. Radiation Consequence of the Stefan-Boltzmann’s Law: T - Temperature at the Surface of the Body T 0 - Temperature of the Environment or the other Body F 1-2 - Shape Factor  - Stefan-Boltzmann Constant  - Emissivity of the Surface of the Body

10. Galerkin Approximation Of The Energy Balance Equation

12. Galerkin Approximation of the Fourrier’s Law:

13. Symmetric Weak Mixed Formulation

14. Finite Element Approximation Function Spaces that Enables Continuity

15. Finite difference time discretization

16. Finite element matrix equations

17. Numerical Examples

18. A Ceramic Strip Model Problem

19. E

20. animacija_straus_vth2.htm

21. A Ceramic Strip Model Problem

23. Transient Temperature Distribution in an Orthotropic Metal Bar

24. 1 2 3 4

25. animacija_ansys_vm113.htm Transient Temperature Distribution in an Orthotropic Metal Bar

27. Steel Ball Numerical Example

32. A Cylindrical Concrete Vessel for Storing the Core of a Nuclear Reactor The walls of the cylinder have tubular cooling vents, which carry a cooling fluid. Heat flow rate through the walls over a period of 5 hours.

33. Nuclear Reactor – Straus7 Non averaged Results, t=62000s

34. Nuclear Reactor – Straus7 Results

35. Nuclear Reactor – Present Results

36. Conclusion A new robust and reliable finite element procedure for calculations of heat transient problem of a solid bodies is presented Approach is fully 3d thus enabling possible bridging with nano and micro analysis of regions of interest in the solid body Reliable semi-coupling with mechanical analysis is enabled also, which is matter of future report

37. ADENDUM Time Integration Schemes

43. Explicit and implicit schemes Explicit scheme: Fully implicit scheme: Crank-Nicholson scheme: Galerkin scheme: