Popular numerical techniques for solving the Boltzmann transport equation (BTE) for sub-micron thermal conduction include the discrete ordinates method and the finite volume method. However, the finite wave speed associated with the BTE can cause large errors in the prediction of the equivalent temperature unless fine angular discretizations are used, particularly at low acoustic thicknesses. In this paper, we combine a ray-tracing technique with the finite volume method to substantially improve the predictive accuracy of the finite volume method. The phonon intensity is decomposed into ballistic and in-scattering components. The former is solved using a ray tracing scheme, accounting for finite wave speed; the latter is solved using an unstructured finite volume method. Comparisons between this new technique and traditional finite volume formulations are presented for a range of acoustic thicknesses, and substantial improvement is demonstrated.

Issue Section:
Microscale Heat Transfer
Keywords:
heat conduction, Boltzmann equation, finite volume methods, ray tracing, phonons, Conduction, Heat Transfer, Microscale, Nanoscale, Numerical Methods
Topics:
Acoustics, Finite volume methods, Heat conduction, Phonons, Ray tracing, Temperature, Boundary-value problems, Errors, Numerical analysis, Electromagnetic scattering, Radiation scattering, Scattering (Physics)
1.
Majumdar
,
A.
,
1993
, “
Microscale Heat Conduction in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
, pp.
7
–
16
.
2.
Joshi
,
A.
, and
Majumdar
,
A.
,
1993
, “
Transient Ballistic and Diffusive Phonon Heat Transport in Thin Films
,”
J. Appl. Phys.
,
74
, pp.
31
–
39
.
3.
Kumar
,
S.
,
Majumdar
,
A.
, and
Tien
,
C.
,
1990
, “
The Differential Discrete-Ordinates Method for Solutions of the Equation of Radiative Transfer
,”
ASME J. Heat Transfer
,
112
, pp.
424
–
429
.
4.
Modest, M. F., 1993, Radiative Heat Transfer, Series in Mechanical Engineering, McGraw Hill, New York, NY.
5.
Sverdrup, P., Banerjee, K., Dai, K., Shih, W., Dutton, R., and Goodson, K., 2000, “Sub-Contiuum Simulations of Deep Sub-Micron Devices under ESD Conditions,” Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices, IEEE, Piscataway, NJ., pp. 54–57.
6.
Narumanchi, S. V. J., Murthy, J. Y., and Amon, C. H., 2003, “Simulation of Unsteady Small Heat Source Effects in Sub-Micron Heat Conduction,” ASME Journal of Heat Transfer, to appear.
7.
Chai
,
J.
,
Lee
,
H.
, and
Patankar
,
S.
,
1993
, “
Ray Effects and False Scattering in the Discrete Ordinates Method
,”
Numer. Heat Transfer, Part B
,
24
, pp.
373
–
389
.
8.
Coelho
,
P.
,
2002
, “
The Role of Ray Effects and False Scattering on the Accuracy of the Standard and Modified Discrete Ordinates Methods
,”
J. Quant. Spectrosc. Radiat. Transf.
,
73
, pp.
231
–
238
.
9.
Laney, C., 1998, Computational Gas Dynamics, Cambridge University Press, Cambridge, U.K.
10.
Murthy
,
J. Y.
, and
Mathur
,
S. R.
,
2002
, “
Computation of Sub-Micron Thermal Transport Using an Unstructured Finite Volume Method
,”
ASME Journal of Heat Transfer
,
124
(
6
), pp.
1176
–
1181
.
11.
Ramankutty
,
M.
, and
Crosbie
,
A.
,
1997
, “
Modified Discrete Ordinates Solution of Radiative Transfer in Two-Dimensional Rectangular Enclosures
,”
J. Quant. Spectrosc. Radiat. Transf.
,
57
(
1
), pp.
107
–
140
.
12.
Sakami
,
M.
, and
Charette
,
A.
,
2000
, “
Application of a Modified Discrete Ordinates Method to Two-Dimensional Enclosures of Irregular Geometry
,”
J. Quant. Spectrosc. Radiat. Transf.
,
64
, pp.
275
–
298
.
13.
Sakami
,
M.
,
Kasmi
,
A. E.
, and
Charette
,
A.
,
2001
, “
Analysis of Radiative Heat Transfer in Complex Two-Dimensional Enclosures with Obstacles Using Modified Discrete Ordinates Method
,”
ASME J. Heat Transfer
,
123
, pp.
892
–
900
.
14.
Olfe
,
D.
,
1970
, “
Radiative Equilibrium of a Gray Medium Bounded by Non-Isothermal Walls
,”
Prog. Astronaut. Aeronaut.
,
23
, pp.
295
–
317
.
15.
Modest
,
M.
,
1989
, “
The Modified Differential Approximation for Radiative Transfer in General Three Dimensional Media
,”
J. Thermophys. Heat Transfer
,
3
(
3
), pp.
283
–
288
.
16.
Chen
,
G.
,
2001
, “
Ballistic-Diffusive Heat Conduction Equations
,”
Phys. Rev. Lett.
,
86
(
11
), pp.
2297
–
2300
.
17.
Sverdrup
,
P.
,
Ju
,
Y.
, and
Goodson
,
K.
,
1998
, “
Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Devices
,”
ASME J. Heat Transfer
,
120
, pp.
30
–
36
.
18.
Murthy
,
J.
, and
Mathur
,
S.
,
1998
, “
Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes
,”
J. Thermophys. Heat Transfer
,
12
(
3
), pp.
313
–
321
.
19.
Leonard
,
B.
,
1979
, “
A Stable and Accurate Convective Modeling Procedure Based on Quadratic Upstream Interpolation
,”
Comput. Methods Appl. Mech. Eng.
,
19
, pp.
59
–
98
.
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