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 Seminar for Rheology, 15 Mart, 2005

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: LUSAS Theory Manual 1, Version 13 STRAUS 7 Verification Manual ANSYS Verification Manual

1 st Law of Thermodynamics Initial condition: Boundary conditions:

Heat Transfer Modes Conduction Convection Radiation

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

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

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

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

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 Shape Factor  - Stefan-Boltzmann Constant  - Emissivity of the Surface of the Body

Galerkin Approximation Of The Energy Balance Equation

Galerkin Approximation of the Fourrier’s Law:

Symmetric Weak Mixed Formulation

Finite Element Approximation Function Spaces that Enables Continuity

Finite difference time discretization

Finite element matrix equations

Numerical Examples

A Ceramic Strip Model Problem

E

animacija_straus_vth2.htm

A Ceramic Strip Model Problem

Transient Temperature Distribution in an Orthotropic Metal Bar

animacija_ansys_vm113.htm Transient Temperature Distribution in an Orthotropic Metal Bar

Steel Ball Numerical Example

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.

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

Nuclear Reactor – Straus7 Results

Nuclear Reactor – Present Results

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

ADENDUM Time Integration Schemes

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