Conference, Proceedings

specimen and heat source R at infinite length l. The model parameters are of the same meaning
like in an ideal model.
This model will be derived in a more detailed way. The dependence of temperature on radius
for the thermal field can be written as ([7], [8])
D−E
+ 2
hc
K r
Δ T ( r)
= −
, (14)
k D(
D − E + 2)
B
where kB is Boltzmann constant, h = h 2π
is modified Planck constant, c is speed of heat
propagation and K, D are fractal measure and fractal dimension, respectively. If the vector r in
2 2 2
2 2 2 2
2 2
the Eq. (14) is expressed like r = rT
− c ( t − t0
) = rT
− c t + 4a0t
− c t0
, where t0 is the time
2
response delay, a 0 = c t0
2 is maximum value of thermal diffusivity (for E = D) and rT is a radius
of fractal space, it is possible (alike by decomposition of the density) to rewrite it to the relation
( D−E
+ 2)
/ 2
( D−E
+ 2)
/ 2 2 2 2 2
Khc
( 4a0t)
⎛ c t0
− r ⎞
T c t
Δ T ( t)
= −
1
B ( 2)
⎜
⎜−
− +
4 0 4 ⎟ . (15)
k D D − E − a t a0
⎝
2 2 2 2
If h = c t0
− rT
is corrected thickness of specimen and the terms in parenthesis can be viewed as
− x
significant in the expansion of exponential function ( 1 − x ≈ e ), we can write
( D−E+
2)
/ 2 ⎡
2 2
Khc
( 4a
− + ⎛ ⎞⎤
0t)
D E 2 h c t
Δ T ( t)
= −
exp⎢−
⎜ +
⎟
⎟⎥
(16)
kB
D(
E − D − 2)
⎢⎣
2 ⎝ 4a0t
4a0
⎠⎥⎦
If we substitute a = 2a0 /( D − E + 2)
for the thermal diffusivity and R = 4a
/ c for the effective
radius of sample (heat source), where a is the coefficient of thermal diffusivity of the body with
fractal dimension D, we obtain
(
[ ( D − E + 2)
2]
D−E
) / 2
2
Khc
( D−E+
2)
2 ⎡ ⎛ h 4at
⎞⎤
Δ T ( t)
=
( 4at)
exp⎢−
⎜ + ⎟ 2 ⎥ . (17)
kB
2D
⎣ ⎝ 4at
R ⎠⎦
The power of heat source hold out to thermal conductivity of real material (characterised by
fractal dimension D) can be written as
P Khc
⎛ D − E + 2 ⎞
= ⎜ ⎟
λ k D ⎝ 2π
⎠
B
( D−E
) / 2
Than we can rewrite Eq. (17) to form
Δ
P ( 4at)
2λ
π
( D−E+
2)
/ 2
T ( t)
= ( E−D
) / 2
⎠
. (18)
2 ⎡ ⎛ h 4at
⎞⎤
exp⎢−
⎜ + ⎟ 2 ⎥ . (19)
⎣ ⎝ 4at
R ⎠⎦
This equation is for R → ∞ (real radius of the specimen and heat source R at infinite length l) is
identical with model at Eq. (12). For E = 3 (3D space) and D = 2 (surface heat source at Eq. (13)
ΔT
( t)
P
2λ
2
4at
⎡ ⎛ h 4at
⎞⎤
exp⎢−
⎜ + ⎟
⎟⎥
. (20)
π ⎣ ⎝ 4at
R ⎠⎦
= 2
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