Boltzmann Transport Equation - COMSOL.com

Details of Solution Procedure
• Original 1-D BTE:
∂I
∂t
∂I
I −
+ µ = o
I
v
, ( µ = cosθ
)
∂x
τ
o
• Nondimensionalize with
t
*
t x
= , η = ,
τ L
o
Kn =
Λ
L
∂I
∂t
*
∂I
+ Kn µ + I
∂η
=
I o
• Split into (+) and (-) directions:
• Discretize angular space at
Gaussian quadrature points:
+
+
⎧∂Ii
∂Ii
⎪ + Kn µ +
*
i
I
⎪
∂t
∂η
⎪
−
−
∂Ii
∂Ii
⎨ + Kn µ + I
*
i
⎪ ∂t
∂η
⎪
+
⎪Dirichlet BCs : Ii
⎩
+
i
−
i
η = 0
= I
= I
o
o
,
,
σ
= T
π
4
( µ > 0)
( µ < 0)
η = 0
i
i
,
I
−
i
η = 1
σ
= T
π
4
η = 1
• After FE run, postprocess:
I
o
( t,
η)
=
1
2
⎡
⎢
⎣
⎡
q(
t,
η)
= 2π
⎢
⎣
n
gp
∑
i=
1
n
2
gp
∑
i=
1
w I
2
i
i
+
i
+
i
+
w µ I
n
+
i
gp
∑
i=
1
+
2
w I
n
i
gp
∑
i=
1
−
i
2
⎤
⎥
⎦
w µ I
i
−
i
−
i
⎤
⎥
⎦
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