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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
  • Illayathambi Kunadian
  • 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
    establishing the temperature gradient
  • ?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
Temp. Grad. Precedence
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
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