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

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Title: 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
• J. M. McDonough
• Ravi Ranjan Kumar
• Department of Mechanical Engineering
• University of Kentucky, Lexington, KY 40506

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

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.

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.

5
Origin of Dual phase lag model
Tzou (1995)
• ?T delay behavior in
• ?q delay behavior in heat-flow
departure

Energy equation
DPL model
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

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

8
Laser heating source term
• J 13.4 Jm?2
• R0.93
• t p96fs
• ? 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
9
Heat Conduction in a solid bar
Initial Conditions
Boundary Conditions
Initial Conditions
Boundary Conditions
10
Discretization and analysis of unsplit DPL
equation
Trapezoidal integration
515111
11
Stability analysis
12
Stability analysis
Von Neumann necessary condition for Stability
13
Stability analysis
Distribution of
14
Finite difference scheme 3D
515111
15
Finite difference scheme 3D
Trapezoidal integration
515111
16
Finite difference scheme 3D
17
Finite difference scheme 3D
18
Finite difference scheme 3D
Douglas-Gunn time-splitting
delta-form Douglas-Gunn time-splitting
19
Results
Z 0.01
Z 0
Hyperbolic
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
21
Results
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
23
Results
DPL
Parabolic
DPL
Parabolic
DPL
Parabolic
Parabolic
Hyperbolic
Parabolic
DPL
DPL
Parabolic
Hyperbolic
Temperature distribution at top surface of gold
film predicted by different models
24
Results
At t 0.3 ps
DPL
Parabolic
Hyperbolic
DPL
Parabolic
Hyperbolic
DPL
Parabolic
Hyperbolic
At t 0.9 ps
Parabolic
Hyperbolic
DPL
DPL
Parabolic
Hyperbolic
Temperature distribution at top surface of gold
film predicted by different models
25
Temperature distribution cont.
At t 1.56 ps
DPL
Parabolic
Hyperbolic
DPL
Parabolic
At t 2.23 ps
Parabolic
Hyperbolic
DPL
Parabolic
Temperature distribution at top surface of gold
film predicted by different models
26
Performance comparison
Dai and Nassar
Present
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
• ?q is responsible for presence of sharp wave
front in heat propagation in CHE conduction
• ?T diminishes the sharp wave front and extends
heat affected zone deeper into the medium

28
Summary and conclusions
• Numerical scheme for solving 3-D DPL equation
• The new numerical formulation of discretizing
DPL directly outperforms Dais 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 (N21). N gt 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