A Simple Numerical Approach For Solving A Dual-Phase-Lag Micro scale Heat Transport Equation

1. A Simple Numerical Approach For Solving A Dual-Phase-Lag Micro scale Heat Transport Equation Illayathambi Kunadian J. M. McDonough Ravi Ranjan Kumar Department of Mechanical Engineering University of Kentucky, Lexington, KY 40506 Advanced Computational Fluid Dynamics

2. Overview of this talk • Introduction • Brief review of origins of DPL equation • Discretization and analysis of unsplit 1-D DPL equation • Stability analysis • Numerical scheme for solving 3-D DPL equation • Computed results from selected problems • 1-D problem • 3-D problem • Summary and conclusions Advanced Computational Fluid Dynamics

3. Challenges in nanoscale heat transfer • From a microscopic point of view, ultrafast laser heating of metals is composed of three processes: • deposition of radiation energy on electrons • transport of energy by electrons • heating of the material lattice through phonon-electron interactions. • During a relatively slow heating process, the deposition of radiation energy can be assumed to be instantaneous and can be modeled by Fourier conduction; but applicability of this approach to very short-pulse laser applications becomes questionable. • We must look for non-Fourier models because the laser pulse duration is shorter than the thermalization time (time required for the phonons and electrons to come into thermal equilibrium) and relaxation time of the energy carriers. Advanced Computational Fluid Dynamics

4. Challenges in nanoscale heat transfer • An alternative is the hyperbolic heat conduction model; but this suffers from violation of the second law of thermodynamics, and physically unrealistic solutions are therefore unavoidable. • Successful attempts to model microscale heat transfer have been made by Qui and Tien, but when investigating macroscopic effects a different model is required. • Tzou proposed the dual phase lag model that reduces to parabolc, hyperbolic, phonon-electron inteaction and pure phonon scattering models under special values of relaxation times. Advanced Computational Fluid Dynamics

5. Origin of Dual phase lag model Tzou (1995) T ~ delay behavior in establishing the temperature gradient q~ delay behavior in heat-flow departure Energy equation DPL model Advanced Computational Fluid Dynamics

6. Numerical Methods • Explicit Methods - 3D • Dai and Nassar developed implicit finite difference scheme • Split DPL equation into system of 2 equations and individual equations solved using Crank-Nicolson scheme and solved sequentially • Discrete energy method to show unconditional stability of numerical scheme • Zhang and Zhao employed iterative techniques like Gauss-Seidel, SOR, CG, PCG to solve 3-D DPL equation • Used Dirichlet conditions, but applying Neumann boundary conditions result in non-symmetric seven banded positive semi-definite matirces not suitable for iterative methods like CG and PCG Advanced Computational Fluid Dynamics

7. Origin of Dual phase lag model • Present method • Formulation based on unsplit DPL equation • Stability shown using von Neumann stability analysis • Extend to 3D • Douglas –Gunn time splitting and delta-form Douglas Gunn time Splitting • Performance compared with numerical techniques available in literature • Results from specific problems Advanced Computational Fluid Dynamics

8. Laser heating source term • J = 13.4 Jm2 • R=0.93 • t p=96fs •  = 15.3nm •  = 1.2104m2s1 • q= 8.5ps • T = 90ps • k = 315Wm1K1 Gold Film Laser Tzou (1995) Intensity of laser absorption L = 100 nm (Intensity of laser ) Qui and Tien (1992) (Intensity of laser ) 3-D laser source Advanced Computational Fluid Dynamics

9. Heat Conduction in a solid bar Initial Conditions BoundaryConditions Initial Conditions BoundaryConditions Advanced Computational Fluid Dynamics

10. Discretization and analysis of unsplit DPL equation Trapezoidal integration 51×51×11 Advanced Computational Fluid Dynamics

11. Stability analysis Advanced Computational Fluid Dynamics

12. Stability analysis Von Neumann necessary condition for Stability Advanced Computational Fluid Dynamics

13. Stability analysis Distribution of Advanced Computational Fluid Dynamics

14. Finite difference scheme 3D 51×51×11 Advanced Computational Fluid Dynamics

15. Finite difference scheme 3D Trapezoidal integration 51×51×11 Advanced Computational Fluid Dynamics

16. Finite difference scheme 3D Advanced Computational Fluid Dynamics

17. Finite difference scheme 3D Advanced Computational Fluid Dynamics

18. Finite difference scheme 3D Douglas-Gunn time-splitting delta-form Douglas-Gunn time-splitting Advanced Computational Fluid Dynamics

19. Results Z = 0.01 Z = 0 Temp. Grad. Precedence Hyperbolic Advanced Computational Fluid Dynamics

20. Results Z = 100 Z = 1 Heat flux. Precedence Parabolic Short pulse laser heating on thin metal film1-D Short pulse laser heating on thin metal film1-D Short pulse laser heating on thin metal film1-D Advanced Computational Fluid Dynamics

21. Results Advanced Computational Fluid Dynamics

22. 3-D Schematic of femtosecond laser heating of gold film 200nm laser beam Work piece-Gold 250nm 500nm 250nm 500nm 100nm 500nm 500nm 500nm 3-D schematic of laser heating of gold film at different locations Advanced Computational Fluid Dynamics

23. Results DPL Parabolic Parabolic DPL DPL Parabolic Parabolic Hyperbolic Parabolic DPL DPL Parabolic Hyperbolic Temperature distribution at top surface of gold film predicted by different models Advanced Computational Fluid Dynamics

24. Results At t = 0.3 ps DPL Parabolic Hyperbolic Parabolic Hyperbolic DPL DPL Parabolic Hyperbolic At t = 0.9 ps Hyperbolic Parabolic DPL DPL Parabolic Hyperbolic Temperature distribution at top surface of gold film predicted by different models Advanced Computational Fluid Dynamics

25. Temperature distribution cont. At t = 1.56 ps DPL Parabolic Hyperbolic DPL Parabolic At t = 2.23 ps Hyperbolic Parabolic DPL Parabolic Temperature distribution at top surface of gold film predicted by different models Advanced Computational Fluid Dynamics

26. Performance comparison Dai and Nassar Present Advanced Computational Fluid Dynamics

27. Summary and conclusions • New numerical technique to solve DPL implicitly • Unconditionally stable numerical scheme for solving 1-D DPL equation • Solves one equation instead of splitting DPL equation into 2 equations and apply discretization • Reduces number of arithmetic operations involved • Reduces computational time • New formulation satisfies von Neumann necessary condition for stability • Heat conduction in a solid bar Semi-infinite slab – temperature raised at one end • qis responsible for presence of sharp wave front in heat propagation in CHE conduction • Tdiminishes the sharp wave front and extends heat affected zone deeper into the medium Advanced Computational Fluid Dynamics

28. Summary and conclusions • Numerical scheme for solving 3-D DPL equation • The new numerical formulation of discretizing DPL directly outperforms Dai’s method of splitting DPL into two equations and then apply discretization • Delta-form Douglas-Gunn time-splitting method outperforms all other numerical techniques – CPU time taken for entire simulation • Explicit method good for small N (N=21). N > 21 all implicit methods except Gauss-Seidel method perform better than explicit method. • CV wave and diffusion models predict higher temperature level in heat affected zone than the DPL model, but penetration depth is much shorter - formation of thermally undisturbed zone. • DPL model - Heat affected zone is significantly larger than other models • Also, DPL results in 3D exhibit similar behavior as the one-dimensional results Advanced Computational Fluid Dynamics