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Non-destructive testing of a building wall by studying natural thermal signals
Energy and Buildings 34 (2002) 63±69
Non-destructive testing of a building wall by studying natural thermal signals D. Defer*, J. Shen, S. Lassue, B. Duthoit Laboratoire d'Artois de MeÂcanique et Habitat, LAMH Faculte des Sciences AppliqueÂes, Universite d'Artois, Technoparc Futura, 62400 Bethune Cedex, France Received 10 March 2001; accepted 16 May 2001
Abstract The behaviour of civil engineering works (structures, buildings, dams, etc.) in time is a current problem which is the subject of deep consideration and numerous research projects. These studies Ð which are aimed at adopting a better approach to repair, maintenance and reinforcement operations Ð have revealed a signi®cant need for the development of means to diagnose and monitor structures. Many nondestructive testing techniques already exist but a major dif®culty in applying them arises from the fact that they are not universal. It is therefore necessary to analyse their limits and de®ne ®elds of application. Choosing a suitable technique is always a delicate process. In addition, the results obtained are generally affected by a considerable degree of uncertainty; cross tests using different techniques make it possible to improve the quality of the diagnosis. Thermal approaches are currently emerging and being developed quickly. They are typically based on infrared thermography measurements. These techniques involve a contact-free analysis and provide overall information on the structure. They are adapted to a qualitative type of research in which the prime objective is to highlight anomalies. However, it is generally complicated and dif®cult to make a quantitative interpretation of the results [1]. This article presents a new thermal method based on the concept of thermal impedance, which can be measured at the surface of a structure. It is adapted to a local quantitative analysis and should be used as a complement to the overall measurements taken by infrared thermography to quantify and re®ne the analysis. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Natural thermal signals; Non-destructive testing; Thermal impedance; Thermal quadripole
1. Theoretical aspects The study described here is based on the use of thermal impedance as a means for characterising a phenomenon. Thermal impedance de®nes a relation between temperature and ¯ux density in a plane. This variable has been used extensively in the case of two-dimensional systems [2], but is used here to characterise one-directional systems. A conventional way of introducing thermal impedance is to use the quadripole theory. 1.1. Thermal quadripole The notion of a quadripole is derived from the simultaneous solving of the heat equation and Fourier law in the *
Corresponding author. Tel.: 33-3-21-63-71-55; fax: 33-3-21-63-71-23. E-mail addresses: [email protected] (D. Defer), [email protected] (J. Shen), [email protected] (S. Lassue), [email protected] (B. Duthoit).
case of a homogeneous medium during one-directional conduction. In such cases, it may be demonstrated that the temperature and ¯ux density thermal state vectors de®ned in two planes perpendicular to the direction of conductive exchanges and at a distance e from one another are linked by a matrix ye M11 M12 ys (1) fe M21 M22 fs s with
q q 1 p M11 cosh e jo sinh e jo ; M 12 a a b jo q q p jo M21 b jo sinh e jo a ; M22 cosh e a (2)
cosh and sinh are, respectively, the cosine and hyperbolic sine and w the pulse. This matrix written in the frequency space is calculated here for two planes at a distance e from one another and made of a homogenous material
0378-7788/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 8 8 ( 0 1 ) 0 0 0 8 6 - X
64
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
Nomenclature a b e f F H [M] Mii pi R Sp i Sxy t T Te x Z
thermal diffusivity (m2 s 1) thermal effusivity (J K 1 m 1 s 1/2) thickness (m) frequency (Hz) flux density (W m 2) transfert function transfert matrix terms of the transfer matrix impedance parameter contact resistance (K m2 W 1) function of sensitivity to the parameter p flux density power spectrum ((W m 2)2 Hz 1) time (s) temperature (8C) sampling rate (s) thickness of the first layer (m) thermal impedance (K m2 W 1 Hz 1)
Greek letters y Fourier transform of heat flux (K) f Fourier transform of temperature (W m 2) o pulsation (rad s 1) Subscripts i input o output characterised by its effusivity e and diffusivity a. If this matrix connects the input and output vectors of a wall, it is referred to as the system transfer matrix. This formalism is not limited to homogeneous materials. In the case of a multilayer medium, the transfer matrix is obtained by cascading the products of elementary matrices associated with each homogeneous wall. 1.2. Thermal impedance A thermal quadripole links the input and output thermal state vectors of a system. It is possible to de®ne the system's input impedance if there is a known relation between the output values [3] Zo o
yo o fo o
(3)
In this case, it is possible to calculate the system's input impedance Zi which is a complex function of the frequency. Zi o
yi o M11 Zs oM12 fi o M21 Zs oM22
(4)
The Mii correspond to the terms of the overall matrix of the system in question.
Fig. 1. Wall tested.
1.3. Definition of the problem A classical system is used here to illustrate and validate the procedure. The three-layer wall is represented on Fig. 1. It consists of two layers of solid concrete 14 and 10 cm thick separated by a 4-cm thick insulating layer. In the experimental scenario, the internal structure of the wall is assumed to be unknown. The ``zero-state'' hypothesis, therefore involves assuming that the wall is made of solid concrete. In this case, the expected impedance is known and comparing it with the experimental impedance should reveal any anomaly. If an anomaly is detected, an inverse parameter identi®cation method should enable a quantitative diagnosis to be made by characterising the discontinuity thermally, and in particular by positioning it within the wall. The thermal impedance is de®ned in the frequency domain on the basis of stresses that may have a random character. The spectral domain used in the tests depends on the depth of the material that is being investigated. The present work seeks to exploit the information contained in the natural energy interactions between the wall and its micro-climatic environment in the absence of any arti®cial source. The outer surface of the wall will be considered as the face providing access to the system. 1.4. Thermal impedance of a three-layer system Quantitative analysis of the results involves developing a theoretical model, on the basis of which it will be possible to identify the required parameters by calibration with the experimental results, as part of an inverse process. The experimental results will indicate a resistive-type anomaly. The calibration model adopted is therefore that of a system consisting of a homogeneous matrix including a localised pure resistance. The value and position of the resistance are unknown. The transfer matrix of such a system is obtained by the cascaded product of the matrices relating to each elementary layer. The ®rst element to be taken into consideration is the homogeneous layer of concrete of thickness x (14 cm in the present case). The above matrix may be associated with this.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
The second is a layer of insulating material (pure resistance) that is characterised by a transfer matrix 1 R (5) Mpoly 0 1 where R epoly =lpoly represents the pure resistance. The ®nal component of the system is a concrete wall 10 cm thick. This part could be characterised by a transfer matrix. However, in this work, it was represented in the form of signals measured over a period of several days. The lowest frequencies taken into account were of the order of 10 5 Hz. These components are already ®ltered by the ®rst two layers. Modelling showed that, in this case, the last layer could be assimilated to a semi-in®nite medium. There is thus a relation between the ¯ux density and output temperature of the polystyrene that depends only on the effusivity b of the concrete. Zo o
yo o 1 p fo o b jo
(6)
Using this relation, it is possible to write a relation between the input variables. The wall input impedance is defined by [4] q 0 1 p jox2 1 tanh Rb jo a C 1 B C (7) Zi o p B q A p 2 b jo @ jox 1 Rb jo 1 tanh a Four parameters appear in this relation. Two of them are characteristic of the concrete, namely the effusivity b and diffusivity a. The other two characterise the discontinuity in terms of its position x and thermal resistance R. A study to determine sensitivity to the various parameters appears to be essential as part of this inverse method identification procedure. This analysis has a two-fold aim, first to study
65
correlations between the parameters in order to define those that can be identified simultaneously, and second to analyse the extent of sensitivity to each identifiable parameter in the range of working frequencies in order to ensure sufficient precision. 1.5. Sensitivity study A sensitivity function was de®ned for each parameter in order to perform a sensitivity study. This function was de®ned as the ratio between the relative variation in impedance to the relative variation in the parameter. Impedance is a complex function of frequency. This discussion will be limited to a study of the modulus, as the conclusions drawn from the study of phase sensitivity functions are identical, in this particular case. The modulus sensitivity functions for each parameter are de®ned by Spi f
DjZ p1 ; p2 ; . . . ; pn ; f j=jZ p1 ; p2 ; . . . ; pn ; f j Dpi =pi
(8)
D|Z| represents the variation of the input impedance modulus Zi following a variation Dpi in parameter pi. To calculate sensitivity functions, it is essential to fix an order of magnitude for each parameter. This requirement does not conflict with the fact that the parameters to be identified are initially unknown, as the sensitivity functions are only used qualitatively. They are used to determine which parameters have a significant influence on the impedance and which parameters can be identified in the frequency band considered in this study. Only an order of magnitude is required for the parameters. Fig. 2 represents the four sensitivity curves in the range [10 3; 3 10 6 Hz]. Comparative study of these curves shows that the impedance is highly sensitive to the effusivity value b of the homogeneous matrix. There is less sensitivity
Fig. 2. Change in sensitivity of the impedance modulus to the various parameters.
66
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
Fig. 3. Sensitivity to position as a function of sensitivity to diffusivity.
to the resistance value R in this range, which means that the estimation of resistance will be less accurate. Sensitivity to the position x and diffusivity a are quite important but change in a similar manner. Fig. 3 represents the change in sensitivity to x as a function of sensitivity to a. This type of representation produces a straight line passing through the origin, thus, showing that the two types of sensitivity are proportional. Simultaneous identi®cation of the two parameters is not possible. These a and x are contained in the impedance expression in the form of the ratio x2/a, and therefore, only this pair can be determined. When non-destructive testing is being carried out, it may be assumed that the diffusivity a of the material is known. The fact that x and a are linked in the term x2/a makes this assumption less detrimental as an error in a would produce a smaller error in x. This rest of this discussion will focus on a situation in which a is known. A typical value, a 6 10 7 m2 s 1, was chosen for the concrete. For simultaneous identi®cation of b, R and x, it is necessary to ®t a non-linear function of these parameters to the experimental function. An iterative procedure was used to do this,inordertominimisetheleastsquareerror.TheLevenberg± Marquardt algorithm was chosen for this purpose [5]. 2. Calculation of experimental impedance This principle adopted in the study involved considering the wall as a linear system that does not vary in time [6]. The system is excited by a ¯ux density stress that produces a change in surface temperature as a response. The numerical ®lter theory indicates that the samples of the two signals are linked by a linear relation representing the discrete linear model. T k
p X i1
ai T k
i
q X i0
bi F k
i
(9)
This equation illustrates the fact that the temperature value at a given instant depends on the past and present values of the flux density excitation and on past changes in temperature. To obtain a frequential representation of the system, the formalism of the z transform is used. The z transform of the temperature sequence is de®ned as y(z) and that of the ¯ux density as f(z) by y z
1 X
T kz k et
f z
1
1 X
F kz
k
(10)
1
On the basis of the time equation linking the input and output signals of the linear system, an equivalent equation linking the various z transforms may be written as follows y z a1 z 1 y z ap z p y z b0 f z b1 z 1 f z bq z q f z
(11)
or y z 1 a 1 z 1 ap z f z b0 b1 z 1 bq z
p q
H z
(12)
H(z) is called the z transfer function of the discrete-time linear system. The z transform is obtained by the relation proposed above. If this transform is calculated for z ejoTe the expression for the discrete Fourier transform is obtained Y o
1 X
y kTe
jokTe
(13)
1
where Te represents the sampling rate. The transfer function determined above and calculated for z ejoTe gives a direct value for the input thermal impedance of the system.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
67
3. Instrumentation
4. Application
Calculating impedance involves measuring changes in thermal ¯ux and temperature in the system input plane. A ``tangential gradient'' ¯uxmeter was used in these experiments [7]. This sensor has the basic advantage of being very thin (of the order of 0.2 mm). Hence it does not disturb the measurements in the chosen frequency range. A sensor with an active surface area of 25 cm 25 cm was chosen so that the heterogeneities inherent in the concrete material would be integrated. A thermoelectric cell integrated in the ¯uxmeter provides simultaneous temperature measurements in the access plane. The measurements were taken at constant intervals by a scanning multimeter and the values stored on a microcomputer.
Flux density and temperature values were measured for 150 h (6.25 days) at 900 s intervals. Figs. 4 and 5 show the changes in ¯ux density and temperature measured on the wall input side over a period of about 6 days. These stresses are the result of natural energy interactions between the wall and microclimatic environment. The preponderant role of the day/night cycle can be seen from these two curves. Fig. 6 represents the power spectral density of the ¯ux obtained by Fourier transformation of the autocorrelation function. Generally speaking, the frequency window used in the study is between 6 10 6 Hz and 10 4 Hz. The preponderance of the day/night cycle is clearly visible. Fig. 7 compares changes
Fig. 4. Flux density as a function of time measured on the outer surface of the wall.
Fig. 5. Temperature as a function of time measured on the outer surface of the wall.
68
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
Fig. 6. Power spectral density of flux as a function of frequency.
in impedance moduli from the experiments and those that are characteristic of an assumed homogeneous medium. The changes are very similar in the high frequency range, which is quite normal as in this case the thermal signals concern only, the ®rst layer of concrete which responds as a semi-in®nite medium. Nearer the lower frequencies, the curves diverge quite noticeably. The experimental curve displays excess impedance, which is characteristic of an anomaly in resistance. Once the anomaly has been detected and de®ned, the following stage involves performing a quantitative study using the inverse procedure described in the theoretical discussion. Three parameters are assumed to be unknown: the effusivity b, position x of the resistance and its value R.
Fig. 8 compares the experimental impedance and that of the three-layer model after parameter optimisation. There is excellent correspondence between the two curves over the entire spectral domain. The optimised parameter values are: concrete effusivity: b 1900 J K 1 m 2 s 1/2; resistance: R 0:7 K m2 W 1 (i.e. epoly 0:035 m pour); position: x 0:15 cm.This corresponds well with the real con®guration. The effusivity value is in good agreement with the results given in the literature. The position of the insulating layer is accurately situated. The discontinuity resistance value is obtained with an estimated error of 10±15%. This uncertainty concerning the resistance value was foreseeable given the low sensitivity to the parameter R in the frequency range investigated here.
Fig. 7. Impedance moduli from experiments and for a simulated homogeneous medium as a function of frequency.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
69
Fig. 8. Moduli of measured impedance and impedance optimised as a function of frequency.
5. Conclusion This work showed that thermal impedance could be used as a means of non-destructive testing and that it provides quantitative information on a conductive system. The method has the further advantage of requiring only simple, inexpensive equipment. It is easy to implement and involves simply placing a sensor on the surface of the material. It is important to underline that this type of measurement is particularly suited to in situ applications, as it requires no checking of boundary conditions. Stresses may be natural or applied arti®cially if it is necessary to analyse certain particular frequencies, or if the natural signals are too weak. In practice, this type of monitoring could be used in association with another investigation method, such as an infrared camera, to detect anomalies. The con®guration tested here is relatively demanding for this method. The thermal stresses in fact spread slowly and the thickness of the material means that the observations often take a long time. Another application is currently being studied. This involves detecting and studying widespread discontinuities situated near the skin of structures. Typical examples include areas of concrete that have been weathered by cycles of freezing and thawing or by severe
thermal contrasts. The ®rst results obtained in this work are extremely encouraging. References [1] D. Balageas, D. Boscher, A. Deom, J. Fournier, R. Henry, La thermographie infrarouge: un outil quantitatif aÁ la disposition du thermicien, Revue GeÂneÂrale de Thermique (Editions EuropeÂennes Thermique et Industrie) 322 (1988) 501±510. [2] J.C. Batsale, D. Maillet, A. Degiovanni, Extension de la meÂthode des quadripoÃles thermiques aÁ l'aide de transformations inteÂgrales, calcul de transfert thermique au travers d'un deÂfaut bidimensionnel, J. Heat Mass Transfer 37 (1) (1994) 111±127. [3] D. Defer, E. Antczak, B. Duthoit, The characterization of thermophysical properties by thermal impedance measurements taken under random stimuli taking sensor-induced disturbance into account, Meas. Sci. Technol. 9 (1998) 496±504. [4] E. Delacre, D. Defer, E. Antczak, B. Duthoit, Random pseudo promptings applied to the thermal characterization of a wet porous material, J. Appl. Phys. 6 (1999) 101±108. [5] W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986. [6] J.S. Bendat, A.G. Piersol, Measurements and Analysis of Random Data, Wiley, New York, 1966. [7] P. Herin, P. Thery, Measurements on the thermoelectric properties of thin layers of two metals in electrical contact. Application for designing new heat flow sensors, J. Meas. Sci. Technol. 3 (1993) 158±163.
Non-destructive testing of a building wall by studying natural thermal signals D. Defer*, J. Shen, S. Lassue, B. Duthoit Laboratoire d'Artois de MeÂcanique et Habitat, LAMH Faculte des Sciences AppliqueÂes, Universite d'Artois, Technoparc Futura, 62400 Bethune Cedex, France Received 10 March 2001; accepted 16 May 2001
Abstract The behaviour of civil engineering works (structures, buildings, dams, etc.) in time is a current problem which is the subject of deep consideration and numerous research projects. These studies Ð which are aimed at adopting a better approach to repair, maintenance and reinforcement operations Ð have revealed a signi®cant need for the development of means to diagnose and monitor structures. Many nondestructive testing techniques already exist but a major dif®culty in applying them arises from the fact that they are not universal. It is therefore necessary to analyse their limits and de®ne ®elds of application. Choosing a suitable technique is always a delicate process. In addition, the results obtained are generally affected by a considerable degree of uncertainty; cross tests using different techniques make it possible to improve the quality of the diagnosis. Thermal approaches are currently emerging and being developed quickly. They are typically based on infrared thermography measurements. These techniques involve a contact-free analysis and provide overall information on the structure. They are adapted to a qualitative type of research in which the prime objective is to highlight anomalies. However, it is generally complicated and dif®cult to make a quantitative interpretation of the results [1]. This article presents a new thermal method based on the concept of thermal impedance, which can be measured at the surface of a structure. It is adapted to a local quantitative analysis and should be used as a complement to the overall measurements taken by infrared thermography to quantify and re®ne the analysis. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Natural thermal signals; Non-destructive testing; Thermal impedance; Thermal quadripole
1. Theoretical aspects The study described here is based on the use of thermal impedance as a means for characterising a phenomenon. Thermal impedance de®nes a relation between temperature and ¯ux density in a plane. This variable has been used extensively in the case of two-dimensional systems [2], but is used here to characterise one-directional systems. A conventional way of introducing thermal impedance is to use the quadripole theory. 1.1. Thermal quadripole The notion of a quadripole is derived from the simultaneous solving of the heat equation and Fourier law in the *
Corresponding author. Tel.: 33-3-21-63-71-55; fax: 33-3-21-63-71-23. E-mail addresses: [email protected] (D. Defer), [email protected] (J. Shen), [email protected] (S. Lassue), [email protected] (B. Duthoit).
case of a homogeneous medium during one-directional conduction. In such cases, it may be demonstrated that the temperature and ¯ux density thermal state vectors de®ned in two planes perpendicular to the direction of conductive exchanges and at a distance e from one another are linked by a matrix ye M11 M12 ys (1) fe M21 M22 fs s with
q q 1 p M11 cosh e jo sinh e jo ; M 12 a a b jo q q p jo M21 b jo sinh e jo a ; M22 cosh e a (2)
cosh and sinh are, respectively, the cosine and hyperbolic sine and w the pulse. This matrix written in the frequency space is calculated here for two planes at a distance e from one another and made of a homogenous material
0378-7788/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 8 8 ( 0 1 ) 0 0 0 8 6 - X
64
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
Nomenclature a b e f F H [M] Mii pi R Sp i Sxy t T Te x Z
thermal diffusivity (m2 s 1) thermal effusivity (J K 1 m 1 s 1/2) thickness (m) frequency (Hz) flux density (W m 2) transfert function transfert matrix terms of the transfer matrix impedance parameter contact resistance (K m2 W 1) function of sensitivity to the parameter p flux density power spectrum ((W m 2)2 Hz 1) time (s) temperature (8C) sampling rate (s) thickness of the first layer (m) thermal impedance (K m2 W 1 Hz 1)
Greek letters y Fourier transform of heat flux (K) f Fourier transform of temperature (W m 2) o pulsation (rad s 1) Subscripts i input o output characterised by its effusivity e and diffusivity a. If this matrix connects the input and output vectors of a wall, it is referred to as the system transfer matrix. This formalism is not limited to homogeneous materials. In the case of a multilayer medium, the transfer matrix is obtained by cascading the products of elementary matrices associated with each homogeneous wall. 1.2. Thermal impedance A thermal quadripole links the input and output thermal state vectors of a system. It is possible to de®ne the system's input impedance if there is a known relation between the output values [3] Zo o
yo o fo o
(3)
In this case, it is possible to calculate the system's input impedance Zi which is a complex function of the frequency. Zi o
yi o M11 Zs oM12 fi o M21 Zs oM22
(4)
The Mii correspond to the terms of the overall matrix of the system in question.
Fig. 1. Wall tested.
1.3. Definition of the problem A classical system is used here to illustrate and validate the procedure. The three-layer wall is represented on Fig. 1. It consists of two layers of solid concrete 14 and 10 cm thick separated by a 4-cm thick insulating layer. In the experimental scenario, the internal structure of the wall is assumed to be unknown. The ``zero-state'' hypothesis, therefore involves assuming that the wall is made of solid concrete. In this case, the expected impedance is known and comparing it with the experimental impedance should reveal any anomaly. If an anomaly is detected, an inverse parameter identi®cation method should enable a quantitative diagnosis to be made by characterising the discontinuity thermally, and in particular by positioning it within the wall. The thermal impedance is de®ned in the frequency domain on the basis of stresses that may have a random character. The spectral domain used in the tests depends on the depth of the material that is being investigated. The present work seeks to exploit the information contained in the natural energy interactions between the wall and its micro-climatic environment in the absence of any arti®cial source. The outer surface of the wall will be considered as the face providing access to the system. 1.4. Thermal impedance of a three-layer system Quantitative analysis of the results involves developing a theoretical model, on the basis of which it will be possible to identify the required parameters by calibration with the experimental results, as part of an inverse process. The experimental results will indicate a resistive-type anomaly. The calibration model adopted is therefore that of a system consisting of a homogeneous matrix including a localised pure resistance. The value and position of the resistance are unknown. The transfer matrix of such a system is obtained by the cascaded product of the matrices relating to each elementary layer. The ®rst element to be taken into consideration is the homogeneous layer of concrete of thickness x (14 cm in the present case). The above matrix may be associated with this.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
The second is a layer of insulating material (pure resistance) that is characterised by a transfer matrix 1 R (5) Mpoly 0 1 where R epoly =lpoly represents the pure resistance. The ®nal component of the system is a concrete wall 10 cm thick. This part could be characterised by a transfer matrix. However, in this work, it was represented in the form of signals measured over a period of several days. The lowest frequencies taken into account were of the order of 10 5 Hz. These components are already ®ltered by the ®rst two layers. Modelling showed that, in this case, the last layer could be assimilated to a semi-in®nite medium. There is thus a relation between the ¯ux density and output temperature of the polystyrene that depends only on the effusivity b of the concrete. Zo o
yo o 1 p fo o b jo
(6)
Using this relation, it is possible to write a relation between the input variables. The wall input impedance is defined by [4] q 0 1 p jox2 1 tanh Rb jo a C 1 B C (7) Zi o p B q A p 2 b jo @ jox 1 Rb jo 1 tanh a Four parameters appear in this relation. Two of them are characteristic of the concrete, namely the effusivity b and diffusivity a. The other two characterise the discontinuity in terms of its position x and thermal resistance R. A study to determine sensitivity to the various parameters appears to be essential as part of this inverse method identification procedure. This analysis has a two-fold aim, first to study
65
correlations between the parameters in order to define those that can be identified simultaneously, and second to analyse the extent of sensitivity to each identifiable parameter in the range of working frequencies in order to ensure sufficient precision. 1.5. Sensitivity study A sensitivity function was de®ned for each parameter in order to perform a sensitivity study. This function was de®ned as the ratio between the relative variation in impedance to the relative variation in the parameter. Impedance is a complex function of frequency. This discussion will be limited to a study of the modulus, as the conclusions drawn from the study of phase sensitivity functions are identical, in this particular case. The modulus sensitivity functions for each parameter are de®ned by Spi f
DjZ p1 ; p2 ; . . . ; pn ; f j=jZ p1 ; p2 ; . . . ; pn ; f j Dpi =pi
(8)
D|Z| represents the variation of the input impedance modulus Zi following a variation Dpi in parameter pi. To calculate sensitivity functions, it is essential to fix an order of magnitude for each parameter. This requirement does not conflict with the fact that the parameters to be identified are initially unknown, as the sensitivity functions are only used qualitatively. They are used to determine which parameters have a significant influence on the impedance and which parameters can be identified in the frequency band considered in this study. Only an order of magnitude is required for the parameters. Fig. 2 represents the four sensitivity curves in the range [10 3; 3 10 6 Hz]. Comparative study of these curves shows that the impedance is highly sensitive to the effusivity value b of the homogeneous matrix. There is less sensitivity
Fig. 2. Change in sensitivity of the impedance modulus to the various parameters.
66
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
Fig. 3. Sensitivity to position as a function of sensitivity to diffusivity.
to the resistance value R in this range, which means that the estimation of resistance will be less accurate. Sensitivity to the position x and diffusivity a are quite important but change in a similar manner. Fig. 3 represents the change in sensitivity to x as a function of sensitivity to a. This type of representation produces a straight line passing through the origin, thus, showing that the two types of sensitivity are proportional. Simultaneous identi®cation of the two parameters is not possible. These a and x are contained in the impedance expression in the form of the ratio x2/a, and therefore, only this pair can be determined. When non-destructive testing is being carried out, it may be assumed that the diffusivity a of the material is known. The fact that x and a are linked in the term x2/a makes this assumption less detrimental as an error in a would produce a smaller error in x. This rest of this discussion will focus on a situation in which a is known. A typical value, a 6 10 7 m2 s 1, was chosen for the concrete. For simultaneous identi®cation of b, R and x, it is necessary to ®t a non-linear function of these parameters to the experimental function. An iterative procedure was used to do this,inordertominimisetheleastsquareerror.TheLevenberg± Marquardt algorithm was chosen for this purpose [5]. 2. Calculation of experimental impedance This principle adopted in the study involved considering the wall as a linear system that does not vary in time [6]. The system is excited by a ¯ux density stress that produces a change in surface temperature as a response. The numerical ®lter theory indicates that the samples of the two signals are linked by a linear relation representing the discrete linear model. T k
p X i1
ai T k
i
q X i0
bi F k
i
(9)
This equation illustrates the fact that the temperature value at a given instant depends on the past and present values of the flux density excitation and on past changes in temperature. To obtain a frequential representation of the system, the formalism of the z transform is used. The z transform of the temperature sequence is de®ned as y(z) and that of the ¯ux density as f(z) by y z
1 X
T kz k et
f z
1
1 X
F kz
k
(10)
1
On the basis of the time equation linking the input and output signals of the linear system, an equivalent equation linking the various z transforms may be written as follows y z a1 z 1 y z ap z p y z b0 f z b1 z 1 f z bq z q f z
(11)
or y z 1 a 1 z 1 ap z f z b0 b1 z 1 bq z
p q
H z
(12)
H(z) is called the z transfer function of the discrete-time linear system. The z transform is obtained by the relation proposed above. If this transform is calculated for z ejoTe the expression for the discrete Fourier transform is obtained Y o
1 X
y kTe
jokTe
(13)
1
where Te represents the sampling rate. The transfer function determined above and calculated for z ejoTe gives a direct value for the input thermal impedance of the system.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
67
3. Instrumentation
4. Application
Calculating impedance involves measuring changes in thermal ¯ux and temperature in the system input plane. A ``tangential gradient'' ¯uxmeter was used in these experiments [7]. This sensor has the basic advantage of being very thin (of the order of 0.2 mm). Hence it does not disturb the measurements in the chosen frequency range. A sensor with an active surface area of 25 cm 25 cm was chosen so that the heterogeneities inherent in the concrete material would be integrated. A thermoelectric cell integrated in the ¯uxmeter provides simultaneous temperature measurements in the access plane. The measurements were taken at constant intervals by a scanning multimeter and the values stored on a microcomputer.
Flux density and temperature values were measured for 150 h (6.25 days) at 900 s intervals. Figs. 4 and 5 show the changes in ¯ux density and temperature measured on the wall input side over a period of about 6 days. These stresses are the result of natural energy interactions between the wall and microclimatic environment. The preponderant role of the day/night cycle can be seen from these two curves. Fig. 6 represents the power spectral density of the ¯ux obtained by Fourier transformation of the autocorrelation function. Generally speaking, the frequency window used in the study is between 6 10 6 Hz and 10 4 Hz. The preponderance of the day/night cycle is clearly visible. Fig. 7 compares changes
Fig. 4. Flux density as a function of time measured on the outer surface of the wall.
Fig. 5. Temperature as a function of time measured on the outer surface of the wall.
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Fig. 6. Power spectral density of flux as a function of frequency.
in impedance moduli from the experiments and those that are characteristic of an assumed homogeneous medium. The changes are very similar in the high frequency range, which is quite normal as in this case the thermal signals concern only, the ®rst layer of concrete which responds as a semi-in®nite medium. Nearer the lower frequencies, the curves diverge quite noticeably. The experimental curve displays excess impedance, which is characteristic of an anomaly in resistance. Once the anomaly has been detected and de®ned, the following stage involves performing a quantitative study using the inverse procedure described in the theoretical discussion. Three parameters are assumed to be unknown: the effusivity b, position x of the resistance and its value R.
Fig. 8 compares the experimental impedance and that of the three-layer model after parameter optimisation. There is excellent correspondence between the two curves over the entire spectral domain. The optimised parameter values are: concrete effusivity: b 1900 J K 1 m 2 s 1/2; resistance: R 0:7 K m2 W 1 (i.e. epoly 0:035 m pour); position: x 0:15 cm.This corresponds well with the real con®guration. The effusivity value is in good agreement with the results given in the literature. The position of the insulating layer is accurately situated. The discontinuity resistance value is obtained with an estimated error of 10±15%. This uncertainty concerning the resistance value was foreseeable given the low sensitivity to the parameter R in the frequency range investigated here.
Fig. 7. Impedance moduli from experiments and for a simulated homogeneous medium as a function of frequency.
D. Defer et al. / Energy and Buildings 34 (2002) 63±69
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Fig. 8. Moduli of measured impedance and impedance optimised as a function of frequency.
5. Conclusion This work showed that thermal impedance could be used as a means of non-destructive testing and that it provides quantitative information on a conductive system. The method has the further advantage of requiring only simple, inexpensive equipment. It is easy to implement and involves simply placing a sensor on the surface of the material. It is important to underline that this type of measurement is particularly suited to in situ applications, as it requires no checking of boundary conditions. Stresses may be natural or applied arti®cially if it is necessary to analyse certain particular frequencies, or if the natural signals are too weak. In practice, this type of monitoring could be used in association with another investigation method, such as an infrared camera, to detect anomalies. The con®guration tested here is relatively demanding for this method. The thermal stresses in fact spread slowly and the thickness of the material means that the observations often take a long time. Another application is currently being studied. This involves detecting and studying widespread discontinuities situated near the skin of structures. Typical examples include areas of concrete that have been weathered by cycles of freezing and thawing or by severe
thermal contrasts. The ®rst results obtained in this work are extremely encouraging. References [1] D. Balageas, D. Boscher, A. Deom, J. Fournier, R. Henry, La thermographie infrarouge: un outil quantitatif aÁ la disposition du thermicien, Revue GeÂneÂrale de Thermique (Editions EuropeÂennes Thermique et Industrie) 322 (1988) 501±510. [2] J.C. Batsale, D. Maillet, A. Degiovanni, Extension de la meÂthode des quadripoÃles thermiques aÁ l'aide de transformations inteÂgrales, calcul de transfert thermique au travers d'un deÂfaut bidimensionnel, J. Heat Mass Transfer 37 (1) (1994) 111±127. [3] D. Defer, E. Antczak, B. Duthoit, The characterization of thermophysical properties by thermal impedance measurements taken under random stimuli taking sensor-induced disturbance into account, Meas. Sci. Technol. 9 (1998) 496±504. [4] E. Delacre, D. Defer, E. Antczak, B. Duthoit, Random pseudo promptings applied to the thermal characterization of a wet porous material, J. Appl. Phys. 6 (1999) 101±108. [5] W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986. [6] J.S. Bendat, A.G. Piersol, Measurements and Analysis of Random Data, Wiley, New York, 1966. [7] P. Herin, P. Thery, Measurements on the thermoelectric properties of thin layers of two metals in electrical contact. Application for designing new heat flow sensors, J. Meas. Sci. Technol. 3 (1993) 158±163.