Conference, Proceedings

Ideal fractal Dirac pulse model [8]
The model which was defined by Eq. (1) can be generalized for heat propagation from any
source type: point (D = 0), linear (D = 1), planar (D = 2) – generally “fractal” with dimension
D ∈ 0,
E ) in one, two, three, generally in E‐dimensional space [8]
202
Δ
Q
T ( t)
= ( E −D
) / 2
cpρ
( 4π
at)
Ideal fractal step wise model
⎟ 2 ⎛ h ⎞
exp ⎜
⎜−
. (7)
⎝ 4at
⎠
The thermal response can then be derived by integration of Eq. (7) for this model on assumption
that the heat power P is constant. The dependence of temperature change on the time can be for
fractal heat source in this case described by
ΔT
( t)
t
2
P
h
∫ exp dt
( E D)
2
0 cp
( 4 at)
4at
⎟ ⎛ ⎞
⎜
⎜−
. (8)
ρ π
⎝ ⎠
= −
By simple deriving (integrating per partes) we can get for s = ( E − D)
2
2 2 t
P ⎡ 1−s
⎛ h ⎞ h
ΔT
( t)
= ⎢t
exp⎜
− ⎟ +
−
∫t
s
cpρ
( 1 s)
( 4π
a)
⎣ ⎝ 4at
⎠ 4a
0
−(
s+
1)
which special case is for s = 1 2 response for planar arrangement (3).
And finally
2 ⎛ h ⎞ ⎤
exp ⎜
⎜−
⎟ d t⎥
, (9)
⎝ 4at
⎠ ⎦
1−s
2
2
1−s
2
1
P ( 4at)
⎡ h h h ⎤
−s
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ P(
4at)
*
ΔT ( t)
=
⎢exp⎜−
⎟ + ⎜ ⎟ Γ ⎜ s,
⎟⎥
=
iΦ
, (10)
s
s
4λ(
1−
s)
π ⎢⎣
⎝ 4at
⎠ ⎝ 4at
⎠ ⎝ 4at
⎠⎥⎦
4λ(
1−
s)
π
where the = c ρ a is thermal conductivity and
iΦ
λ p
2
e
π
Γ ( s,
x )
π
− x
2
∗
= s
2(
1−s
)
− x
s , (11)
t
s −(
s+
1)
−A
t
where x = h 4at
and Γ ( s,
A t)
= A ∫t
e dt
is the upper incomplete gamma function.
0
This equation is for s = ( E − D)
2 and neglecting upper incomplete gamma function equal to
Δ
P ( 4at)
2λ
π
( D−E
+ 2)
/ 2
T ( t)
= ( E−D
) / 2
2 ⎛ h
exp ⎜
⎜−
⎝ 4at
and for E = 3 (3D space) and D = 2 (surface heat source) when s = 1 2 equal to
P
ΔT
( t)
=
2λ
Real fractal step wise model
⎟ ⎞
⎠
⎟ 2
4at
⎛ h ⎞
exp ⎜
⎜−
. (13)
π ⎝ 4at
⎠
The heat loss effect was solved in a new model by defining the initial and boundary conditions
for basic heat transport equation. The heat losses were taken into account at real radius of the
(12)