CONDUCTION IN A DIELECTRIC FILM

ALEJANDRO GUAJARDO-CUELLARADVISOR: DR. MIHIR SEN

acknowledgements :

CONACYT

OUTLINE

• MOTIVATION

• DESCRIPTION OF THE PROBLEM

• MATHEMATICAL MODELS

• SOLUTIONS

• RESULTS

• CONCLUSIONS

• FUTURE WORK

MOTIVATION

• HEAT TRANSFER IN MICRO AND MACRO SCALES CANNOT BE DESCRIBED IN THE SAME WAY AS MACRO SCALE.

• DIFFERENT APLICATIONS CAN BE FOUND WHERE HEAT TRANSFER IN MICRO SCALE IS A FIGURE OF MERIT.

• CARBON NANO TUBES, INTEGRATED CIRCUITS.

SOME SAMPLES

http://www.nec.com/global/corp/H0602.html

http://www.ewels.info/.../nanotubes/tube.angled.jpg

http://www.ipt.arc.nasa.gov/interconnect1.html

PROBLEM PROPOSED

L

T1T0

x T= T0 at t=0

Fourier law

HYPERBOLIC MODEL

DIFUSSION BY RANDOM WALK

Tzou proposed the following equation:

This PDE does not have close solution. Tzou et al proposed an algorithm to solve the equation based in

Laplace Transform. Shiomi and Maruyama studied Non-Fourier heat conduction in a single-walled carbon

nanotube. They use molecular dynamics and the equation (1) and compare the result. Suggesting that

equation (1) describes the phenomenon due they obtain similar results as molecular dynamics

(1)

Results

CONCLUSIONS

• The models give different behavior for the same problem, the steady state solution is reached at different times.

• The same behavior for fourier and random walk is obtained.

• The second derivative in time in the hyperbolic heat equation gives the wave-like behavior meanwhile the first derivative works as a damping of the wave.

• This results give more understanding of the mathematical models proposed.

FUTURE WORK

• EXTEND THE PROBLEM TO 2-D• STUDY WITH MORE DETAIL THE RANDOM

WALK MODEL, AND TRY TO SOLVE THE PROBLEM WITH UNBALANCED PROBABILITY, OR STORAGE OF ENERGY OF THE DEFECT.

• APPLY THE RESULTS TO A SPECIFIC PROBLEM, LIKE CONDUCTION IN A CARBON NANO-TUBE.