Boltzmann Transport Equation - COMSOL.com
Boltzmann Transport Equation - COMSOL.com
Boltzmann Transport Equation - COMSOL.com
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Details of <strong>Boltzmann</strong> <strong>Transport</strong> <strong>Equation</strong> (BTE)<br />
• Phonon intensity:<br />
Iω ( t,<br />
v,<br />
r)<br />
= v ωf<br />
( t,<br />
v,<br />
r)<br />
D(<br />
ω) / 4π<br />
• BTE be<strong>com</strong>es EPRT:<br />
∂I<br />
∂t<br />
ω<br />
+ v ⋅∇I<br />
ω<br />
=<br />
I<br />
ω o<br />
τ<br />
− I<br />
o<br />
ω<br />
,<br />
Iω<br />
o<br />
=<br />
1<br />
∫<br />
I<br />
ω<br />
4π<br />
Ω= 4π<br />
dΩ<br />
*Equilibrium phonon intensity<br />
determined by Bose-Einstein statistics<br />
• For 1-D,<br />
∂I<br />
∂t<br />
ω<br />
∂I<br />
+ v cosθ<br />
∂x<br />
ω<br />
=<br />
I<br />
ω o<br />
τ<br />
− I<br />
o<br />
ω<br />
x<br />
θ<br />
Ι ω<br />
L<br />
• For each angle (θ ), solve non-linear equation with iterations for<br />
– Solving for Ι ω .<br />
– Updating Ι ωο .<br />
• Heat flux:<br />
q<br />
=<br />
Ω=<br />
∫∫<br />
ω<br />
D<br />
Iω<br />
cosθ dω<br />
dΩ<br />
4π<br />
0<br />
• Internal energy:<br />
u<br />
1 ω D<br />
( ≈ cT ) = ∫ ω<br />
f D(<br />
ω)<br />
dω<br />
dΩ = ∫ ∫<br />
4<br />
π<br />
0<br />
Ω= 4π<br />
I<br />
v<br />
ω<br />
dω<br />
dΩ<br />
7
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