Dubravka Mijuca, Bojan Medjo Faculty of Mathematics, Department of Mechanics

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

4. Heat Transfer Modes • Conduction • Convection • Radiation

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

6. Wood 0.05 Water 0.7 Glass 0.8 Steel10-20 Iron 80 Copper 400 Silver 450 Thermal Conductivities 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,T0 – Temperatures of the surface and the Fluid • hC– 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 T0 - Temperature of the Environment or the other Body F1-2 - Shape Factor s - Stefan-Boltzmann Constant e - Emissivity of the Surface of the Body

10. Galerkin Approximation Of The Energy Balance Equation

11. 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. A Ceramic Strip Model Problem E

20. A Ceramic Strip Model Problem animacija_straus_vth2.htm

21. A Ceramic Strip Model Problem

22. A Ceramic Strip Model Problem

23. Transient Temperature Distribution in an Orthotropic Metal Bar

24. Transient Temperature Distribution in an Orthotropic Metal Bar 4 2 3 1

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

26. Transient Temperature Distribution in an Orthotropic Metal Bar

27. Transient Temperature Distribution in an Orthotropic Metal Bar

28. Steel Ball Numerical Example

29. Steel Ball Numerical Example

30. Steel Ball Numerical Example First iteration t=250 Last iteration t=5819

31. Steel Ball Numerical Example

32. Steel Ball Numerical Example

33. 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.

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

35. Nuclear Reactor – Straus7 Results

36. Nuclear Reactor – Present Results

37. 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

38. ADENDUM Time Integration Schemes

39. PRIMAL FORMULATIONS

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