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Воронежский государственный технический университет

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

In both cases, for a semi-infinite medium the transfer ( ) is the same.

4. Responses in the Semi-Infinite Plane

4.1. Asymptotic Behavior Analysis at

At	, (	)	; thus the transfer function			(	) becomes:
				(	)	( )		(16)
which asymptotic behaviors are given by:
			{	(	)	(	)	(17)
			{	(	)	(	)	(17)
				(	)	(	)
The fractional integration behavior of order						between the input flux ̅( ) and the
temperature	̅(	) is at mid-way between an integration behavior of order						which	is
characteristic of a capacitive behavior, and a proportional behavior (integration of order									)
characteristic of a resistive behavior.

4.2. Asymptotic Behavior Analysis at

The analysis of the influence of	(	) on	the transfer function (					) through its
asymptotic behaviors leads to:
		(	)
{			.		/			(18)
(	)		.		/
(	)
Hence,
(	)		(	)
{			.			/	.	(19)
(	)		.			/
(	)

The same asymptotic behavior is noticed at any place of the semi-infinite medium.

4.3. Frequency Response Analysis at

The frequency response (

) is given by:

(

)

( ) (

(20)

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	From the Formal Concept Analysis to the Numerical Simulation …							99

At	, the block (	)	thus the transfer function (				) becomes:
		(	)	(	).			(21)
The gain and the phase of the transfer function ̅(						) are:
		{	\| (	)\|	.			(22)
		{	\| (	)\|	.			(22)
			( (	))
			( (	))
Bode diagrams and Black-Nichols diagram of (						) are represented in (Figure 3) and

(Figure 4), respectively.

4.4. Frequency Response Analysis at

At	, the 3rd block	(		) becomes:
				(	)		.		/		(23)
				(	)						(23)

Knowing that		.	/		.	/	√		√ , the block (		) is written as:
Knowing that		.	/		.	/			√ , the block (		) is written as:
		(		)	.			/	.	/	(24)
					.			/	.	/

thus the gain and the phase of this block are:

	0
	-40		0
(dB)	-40		1
			1
	-80		5
Gain			5

			10
	-120		50
	-120		50
			100
	-160	-7	-6	-5	-4	-3	-2	-1	0	1	2	3
	10		10	10	10	10	10	10	10	10	10	10
	0
(deg)	-45		0
	-45		1
			1
			5
Phase	-90		5
	-90
			10
	-135		50
	-135		50
			100
	-180	-7	-6	-5	-4	-3	-2	-1	0	1	2	3
	10		10	10	10	10	10	10	10	10	10	10
							Frequency (rad/s)

Figure 3. Bode diagrams of (	) for different positions of the temperature sensor in a semi-infinite
aluminum medium.

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

	40
								0
	20							1
	20
								5
	0							10
	0							50
								50
								100
	-20
(dB)	-40
Gain	-40
Gain
	-60
	-80
	-100
	-120
	-270	-225	-180	-135	-90	-45	0	45
				Phase (deg)

Figure 4. Black-Nichols plots of	(	)	for different		positions of the sensor in a			semi-infinite
aluminum medium.
{	\|	(	)\|	.		/
				.		/
						.		(25)
	(	(	))		.		/
	(	(	))		.		/

-At the low frequencies, the behavior of this block is similar to the one observed

when :

{	\| (	)\|
	(	(

-At the medium frequencies, especially frequency:

{\| (	)\|	(
(	(	))

))	.	(26)
))
when		⁄ , the cutoff
	).	(27)

-At the high frequencies, the behavior of this block is like:

{	\|	(	)\|	.	(28)
	(	(	))

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From the Formal Concept Analysis to the Numerical Simulation …	101

Now, we can look deeply into the transfer function:

	̅(	)				.		/	.	/ .			(29)
						.		/	.
			(		)
			(		)
The asymptotic behaviors of		(		) are found to be:
-	At low frequencies:
	{		\|		(		)\|		.				(30)
			\|		(		)\|
					( (		))


The behavior is similar to the one observed when									.
-	At high frequencies:
	\|	(	)\|						.		/
									.		/


	{									.			(31)
	( (		))				(		.			/ )
	( (		))				(		.			/ )

Bode diagrams and Black-Nichols plots in a semi-infinite aluminum medium are merged

in Figure 3 and Figure 4 with various values of		. The fractional integration behavior of
order	is present all over the frequency range for	, it disappears gradually at higher
frequencies when increases.

4.5. Time Response Analysis

When analyzing this system in time domain, a special attention should be taken. In fact, because of the presence of the exponent in the expression of ( ), the inverse Laplace

transfer in	simulation is not always possible when taking into consideration special inputs
̅( ). (Eq.	32) shows the inverse Laplace transform			* + of the output temperature.
	( )	*̅(	)+	* (	) ̅( )+	(32)

Nevertheless, some time domain simulations could be made without using approximations. In fact, three time domain simulations for three well known input types are proposed: the impulse, the step and the sinusoidal inputs.

4.5.1. Impulse Response

The impulse response is obtained by substituting ( ) by its value shown in (Eq. 12) and ̅( ) by * ( )+ . Accordingly, the impulse response is given by:

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

( )

(

√ ⁄ )

√ ⁄

(33)

√

Knowing that (Battaglia, 2008):

					( )
.		√	/		( )	(34)


	√			√
	√			√

where

, the impulse response can be presented as follows:

	( )	( )


		√
		√
	Figure 5 shows the outputs for the impulse at different locations of
	of in a semi-infinite aluminum medium.

(35)

and different values

4.5.2. Step Response

The step ( ) response of magnitude shown in (Eq. 12) and ̅( ) by * ( )+

is obtained by substituting ( ) by its value ⁄ . Accordingly, the step response is given by:

				(	)		(√	√ ⁄
				(	)		(√
	1.4	x 10-5
	1.4
								0
	1.2							1
	1.2							5
								5
								10
(°C)	1							50
(°C)								100
Variation	0.8
Temperature	0.4
	0.6
	0.2
	0	0	10	20	30	40	50	60
				(a):	Time (s), T0=0°C

Figure 5. Impulse responses of (			) for (a)
and (b)	and		.
Knowing that (Battaglia, 2008):

	.		√ /	√
	.	⁄	√ /	√
		⁄

)		.		√ ⁄	/		(36)
				⁄
	1.4	x 10-5
	1.4
							1
	1.2						10
	1.2						20
							20
							40
(°C)	1						80
(°C)
Variation	0.8
Variation
Temperature	0.6
Temperature	0.4
	0.2
	0	0	20	40	60	80	100
			(b): Sensor position x (mm), T0=0°C
						and	;
(	)			. √ /			(37)
				. √ /			(37)

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	From the Formal Concept Analysis to the Numerical Simulation …		103

where	( ) is the complementary error function and	, the step response can be
presented as follows:

	1
			0
	0.9		1
	0.8		5
	0.8		10
			10
(°C)	0.7		50
	0.7		100
			100
Variation	0.6
Variation	0.5
Temperature	0.5
	0.4
	0.3

	0.2
	0.1
	0	0	30	60	90
			(a): Time (s), T0=0°C, Flux=2000W/m2

Figure 6. Step responses of		(	) for (a)
(b)	and		.
	(	)	( √

		1
											1
		0.9									10
											10
		0.8									20
		0.8									40
											40
	(°C)	0.7									80
		0.7

	Variation	0.6
	Variation	0.5
	Temperature	0.5
		0.4
		0.3

		0.2
		0.1
		0	0	25	50	75	100	125	150	175	200
				(b): Sensor position x (mm), T0=0°C, Flux=2000W/m2
								and			; and
(	)				. √	/)					(38)
			√		. √	/)					(38)

Figure 6 shows the outputs for the step function at different locations of			and different
values of in a semi-infinite aluminum medium.
4.5.3. Stationary Harmonic Response
The entry at	is supposed to be
	( )	( ).	(39)

The system being supposed linear, the expression of the output temperature is given by:

(	)	\| (		)\|		.	( (	))/		(40)
then replacing (	) by its value in (Eq. 29)
(	)	.			/	(	(	.	/ ))	(41)
		.			/

Let
			⁄	and			⁄			(42)

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

Then
			(	)	.	/		√	.		√	/		(43)
			(	)	.	/			.		√	/		(43)
	25
														1e-006
	20													1e-005
	20													0.0001
														0.0001
(°C)	15
(°C)
Variation	10
Variation
Temperature	5
Temperature	0
	-5
	-100	1	2	3	4	5	6	7	8	9	10	11	12	13
						Time (hour), T0=0°C, Flux=2000W/m2

Figure 7. Harmonic responses of

(

) for ( ⁄

)

+ and

⁄ .

(

)

(44)

Figure 7 illustrates the outputs for a harmonic

function input for different

values of

( ⁄ ) *

+ and for

⁄

in a semi-infinite aluminum medium.

In order to make the analysis easier, let’s

consider

the

temperature at

as a

reference, thus we can write:

(

)

(

(45)

Hence the temperature at

will be written as:

(

)

√

(

√

(46)

For illustration purposes, let

with

. Figure 8 shows the outputs for

a harmonic function input for different values of

+ in a semi-

infinite aluminum medium.

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From the Formal Concept Analysis to the Numerical Simulation …	105

Figure	8. Harmonic	responses of	(	) when	, for ( )	, ( )	,
( )	, ( )	, ( )	and (	)	.

t	x
0

Figure 9. Finite unidirectional plane of thickness L.

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