Conference, Proceedings

comparison of results we can determine the suitability of studied laminating foil for the
photovoltaic panel application.
2. Models
Classical and fractal approach
Ideal planar Dirac pulse model [1], [3]
The dependence of the temperature change on time for a simple model after a Dirac pulse of
heat supply prom surface heat source can be derived with using of the term [1]
⎟ 2
Q ⎛ h ⎞
ΔT
( t)
=
exp ⎜
⎜−
, (1)
c 4 at
⎝ 4at
pρ
π
⎠
where Q = PΔt
means total pulse heat energy (P is a power of heat source, Δt width of the
pulse), cp is specific heat, a is thermal diffusivity and t is time.
Ideal planar Step wise model [1], [3]
The thermal response for this model can be derived by integration of Eq. (1), assuming that the
heat power P is constant. The dependence of temperature change on the time can be for planar
heat source in this case described by
t
2
P
h
ΔT
( t)
∫ exp dt
0 c 4 at 4a
t ⎟
p
⎟
⎛ ⎞
=
⎜
⎜−
. (2)
ρ π ⎝ ⎠
By a simple deriving (integrating per partes) we can get
or
or
ΔT
( t)
2P
c ρ 4π
a
⎡
⎢
⎣
2 2
⎛ h ⎞ h
t exp ⎜
⎜−
⎟ +
⎝ 4at
⎠ 4a
= ∫ −
t
3 2
p
P 4at
⎡ ⎛ h ⎞ h ⎛ 1 h ⎞⎤
P 4at
ΔT ( t)
⎢exp
Γ ,
iΦ
2λ
π
⎜−
⎥ =
4a
t ⎟ −
⎣ ⎝ ⎠ 4at
⎝ 2 4at
⎠⎦
2λ
0
t
2 ⎛ h ⎞ ⎤
exp ⎜
⎜−
⎟ d t⎥
, (3)
⎝ 4at
⎠ ⎦
2
2
= ⎜ ⎟ ⎜ ⎟
*
, (4)
P 4at
⎡ 1 ⎛ h ⎞ h ⎛ h ⎞⎤
P 4at
ΔT ( t)
⎢
Φ
2λ
π
⎜−
⎥ =
4a
t ⎟
⎢⎣
4at
⎜ 4at
⎟
⎝ ⎠ ⎝ ⎠⎥⎦
2λ
2
*
= exp⎜
⎟ − erfc⎜
⎟ i , (5)
respectivelly. At this equation, the λ = cpρ a is the thermal conductivity, erfc(x) is
complementary error function (also called the Gauss error function, see Appendix). It is defined
as
erfc(
Γ ( 1 2,
x )
∫ ∞
2
2
−x
x) = = e d
π π x
2
x
x
e 2
−
∗
and i = − x erfc(
x)
t
Φ , (6)
π
1 2 −3
2 −A
t
where x = h 4at
and Γ ( 1 2,
A t)
= A ∫t
e dt
is the upper incomplete gamma function.
0
201