106

Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al.

Note that, as long as the frequency of variation of the temperature is low relatively to

the diffusivity , the temperature variation is uniform and harmonic.

When the frequency of variation of the temperature

increases, the diffusion depth of

the temperature decreases in the semi-infinite medium.

thickness , conductivity

, diffusivity , and of initial temperature zero all over the medium

(Figure 9). A heat flux

( ) is normally applied on its surface at

; it is assumed that

there are no losses at this point. Consequently, a temperature gradient (

) appears, which

is a function of the independent time variable as well as the abscissa

of the temperature

measurement point, with

,

- inside the medium.

( )

( )

.

(47)

(

)

.

(48)

(

)

( )

(49)

(

)

.

(50)

( ) as shown in Figure 9.

The initial condition of the temperature being zero, Laplace transform

* + of (Eq. 47) is:

̅( )

̅( )

where ̅( )

* ( )+

(51)

From the Formal Concept Analysis to the Numerical Simulation …

107

̅( )

( ) √ ⁄

( ) √ ⁄ .

(52)

Resolving (Eq. 52) for the boundary conditions (Eq. 49-50) gives the values of

( ) and

( ), let:

( )

( )

̅( )

√

{

(53)

( )

√

( )

√

̅

(

)√ ⁄

( )√ ⁄

( ) √

√ ⁄

√ ⁄

̅( ) .

(55)

function ( ) between the input flux ̅( ) and the output temperature ̅(

) is:

̅(

)

(

)√ ⁄

( )√ ⁄

(

)

,

(56)

̅

( )

√

√ ⁄

√ ⁄

then, introducing the hyperbolic functions

and

the

transfer

function ( )

becomes:

(

)

̅(

)

((

)√ ⁄ )

,

(57)

̅

( )

√

. √ ⁄ /

.

√ ⁄ /

or rewritten as:

(

)

( )

(

) (

) ,

(58)

where:

108

Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al.

̅(

)

̅( )

√

( )

̅(

)

̅(

)

(

)

̅(

)

(59)

̅(

)

(√ ⁄

)

(

)

̅(

)

(√ ⁄

)

̅(

)

(√ ⁄

)

{

with the following considerations:

{

.

/

(60)

(

)

One can derive from (Eq. 60) that

and

are linked by the following relation:

.

/

.

(61)

One can derive also the thermal diffusion time constant defined as:

.

(62)

To clarify, the different temperatures found in the system (Eq. 59) are as follows:

-

̅(

) is the measured temperature at

if the medium is considered to be

a semi-infinite medium.

-

̅(

) is the measured temperature at

of the finite medium of length .

-

̅(

) is the measured temperature at

,

,

-.

Consequently, the transfer function (

) of the system can be seen as four cascading

functions represented in (Figure 10).

The function

depends on the medium physical characteristics, it links the input flux

̅( ) and the fractional derivative of order

of the temperature at

.

Then,

( ) is a fractional integrator of order

that gives the temperature at

. Its

output is the temperature ̅(

).

The third block is the transfer function (

) between the temperatures ̅(

) and

̅(

).

From the Formal Concept Analysis to the Numerical Simulation …

109

The fourth part is the transfer function

(

) having as input ̅(

) and as output

the temperature ̅(

).

H(x, s, L)

(0, s, oo)

(0, s, L)

s0.5T

(0, s, oo)

T

(x, s, L)

T

T

I 0.5(s)

F(0, s L)

G(x, s, L)

H0

Figure 10. Block diagram illustrating the transfer function

(

) of the finite medium.

6. Responses in Finite Plane

6.1. Asymptotic Behavior Analysis at

At

,

(

)

(

)

,

thus

the temperature ̅(

) of

the finite

medium of length

is given by:

̅(

)

( )

(

) ̅( ).

(63)

The transfer function (

) allows the analysis of the influence of the medium finite

character

on the

temperature

at

. As

such, let the change of variable

√ ⁄ .

(

)

( )

(64)

which is a result similar to what was found in a semi-infinite medium at

.

Now consider when

then

, as

(

)

, thus:

(

)

, accordingly it is found that:

√ ⁄

(

)

( )

(65)

(

)

.

(66)

110

Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al.

integrator of order

.

For low

instances, corresponding to

high

frequencies, when

then

, as

(

)

, thus:

(

)

( )

(67)

Accordingly an asymptotic fractional order behavior appears when

between the

For

high

instances,

corresponding

to

low

frequencies,

when

then

(

)

, thus:

(

)

(68)

The same asymptotic behavior applies as at

. It is of capacitive type characterized

by an integrator

of order

; it is different than the

behavior in a

semi-infinite medium

characterized by a fractional integrator of order

.

For

low

instances, corresponding

to

high

frequencies,

when

then

,

as

√ ⁄

,

thus

(

)

; it

is resolved introducing (Eq.

61)

a

√ ⁄

relation between

and

; accordingly:

√

√

(

)

⁄

(69)

then,

(

)

( )

√

(70)

Now that the mathematical model is established, frequency domain analyses (

)

can be done. The resulting transfer function (

) is the concatenation of four cascading

From the Formal Concept Analysis to the Numerical Simulation …

111

the boundary where the flux is applied,

, the transfer function is expressed by:

(

)

(

) (

) (

)

(71)

with (

)

, then

(

) reduces to::

(

)

(

) (

)

(72)

medium are represented in Figure 11.

The frequency response study of the function (

) allows

understanding

the

influence of the finite character of the medium on the temperature at

. Actually Figure

12 represents Bode diagrams of this transfer for three different values of

*

+

in a finite aluminum medium.

Figure 11. Bode diagrams of

(

) in a finite medium at

.

Two asymptotic behaviors clearly appear:

At the low frequencies, a fractional integration behavior of order

prevails. Actually,

, (

)

.

/

{ |

(

)|

.

/

(73)

( (

))

112

Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al.

t the

high frequencies, a proportional

behavior prevails. Actually,

,

(

)

{

| (

)|

(74)

( (

))