Sensors and Actuators A 163 (2010) 441–448
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Alternative fitting procedures to enhance the performance of resistive sensors for thermal transient measurements B.M. Suleiman ∗ Department of Applied Physics, College of Sciences, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates
a r t i c l e
i n f o
Article history: Received 7 January 2010 Received in revised form 30 April 2010 Accepted 8 June 2010 Available online 12 June 2010 Keywords: Computational fitting procedures Dynamic plane source technique Thermal conductivity Thermal diffusivity Linear regression
a b s t r a c t The main concern in most of transient techniques is to obtain a controlled heat flow in a prescribed direction, so that the actual boundary conditions in the experiment agree with those assumed in the theory. Enhancements of the fitting procedures in transient dynamic techniques have been established. One of these dynamic techniques known as the extended dynamic plane source is discussed in details. This technique has been used for simultaneous measurements of thermal conductivity, diffusivity and the specific heat from a single transient recording of the temperature increase. The technique uses a resistive sensor as both, a plane heat source and temperature sensor, i.e. in similar manner as the wire in the transient hot-wire method. The sensor consists of a resistive heater pattern cut from a thin sheet of metal (Ni) covered on both sides with thin layers of an insulating material. The conducting pattern of the sensor, known as the hot-disk configuration, has the shape of spiral strips. Infrared camera images of the sensor were taken, at the beginning of the transient recording, to illustrate the deviations from the ideal conditions due to design defects within the sensor. Alternative fitting procedures have been introduced in order to meet the design criteria of heat and temperature sensor and to fulfil the isothermal boundary conditions required by experimental setup. Within the total time of transient recording, and using a sequence of computational steps, it is possible to find a correct “optimal” time interval to enhance the fitting procedures. These computational steps are based on fitting procedures that provides the selection of the optimal time interval within the total measuring time and thus to obtain more accurate and reliable results. These procedures have been tested using measurements on acrylic “Plexiglas” sample; the corresponding deviations from published data did not exceed 1.5% and 2% in thermal conductivity and diffusivity, respectively. It is another alternative to the standard fitting procedures which mainly based on linear regression and least square fitting. It is anticipated that these fitting procedures have the potential to be used in similar transient techniques. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The transient dynamic techniques for thermal properties measurements can be divided into contact and non-contact (laser flash) techniques. The major two advantages of the contact transient dynamic techniques [1–8] are short measuring time and simple experimental setup. The most common ones are the hotwire and the hot-strip. Each uses a line heat source/sensor (wire or strip) that is embedded in the specimen initially kept at uniform temperature. The principle of these techniques is simple. The sample is initially kept at thermal equilibrium, and then a small disturbance is applied to the sample in a form of a short heating pulse through the sensor. Using these type of sensors, it is possible to measure both the heat input and the temperature
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increase, from which the thermal conductivity or both (only in hot-strip case) and thermal diffusivity a are simultaneously determined. Based on hot-strip technique the transient plane source (TPS) technique was emerged which is characterized by the transient temperature rise of a plane heat source/sensor, known as hot disk [1,8]. Measurements are simply performed by recording the voltage (resistance/temperature) variations across the sensor during the passage of a heating current in a form of a constant electrical pulse. The theory of the TPS method is based on a three-dimensional heat flow inside the sample, which can be regarded as an infinite medium, if the time of the transient recording is ended before the thermal wave reaches the boundaries of the sample. In the dynamic plane source (DPS) technique [2], the hot disk is used as a plane heat source so that the experimental arrangement resembles a one-dimensional heat flow and used for measurements of relatively high conducting materials. The main features distinguishing DPS from the TPS can be summarized as:
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Fig. 1. (Top) An actual picture of the sensor held in a vertical position via 4 screws and Plexiglas support (bottom) schematic drawing of the sample pieces and experimental setup.
(i) For samples with ≥ 2 W m−1 K−1 , the DPS is arranged for a one-dimensional heat flow into a finite sample which is in a contact with relatively poor heat conductor to establish the adiabatic conditions. (ii) DPS works in the time region where the sample is treated as a finite medium and is not restricted only to the time region where the sample is regarded as infinite medium. (iii) Even if the experimental arrangement resembles a onedimensional heat flow, the DPS has the potential to give , a, and Cp from a single transient recording. In spite of these features to get reliable results the data handling should be carried out in such a way that the deviations for the ideal conditions can be justified. In this work alternative fitting procedures is introduced to handle such deviations.
2. The experiment 2.1. Experimental setup There are several factors that affect the reliability of a specific technique to measure thermal properties. Some of these factors are the required accuracy, the speed of operation, the physical nature of material, the geometry of the available sample and the performance under various environmental conditions. However, in most techniques the main concern is to obtain a controlled heat flow in a prescribed direction, so that the actual boundary conditions in the experiment agree with those assumed in the theory. The extended dynamic plane source (EDPS) technique [9] is a modified version of DPS [2] which is also based on a one-dimensional heat flow into a finite sample and used for measurements of low conducting materials. With ≤ 2 W m−1 K−1 , the sample must be kept in contact with very good heat sink such as copper to establish the steady state conditions in relatively short time. An actual picture of the sensor and the configuration of the experimental setup in EDPS are shown in Fig. 1
The plane heat source/sensor (hot disk), which simultaneously serves as the heat source and thermometer is clamped between two identical sample pieces of cylindrical shape. The identical pieces are used to provide symmetry to the heat flow through the samples and into a heat sink (high thermal conducting material), copper in this case. Using a better heat sink leads to better experimental conditions and enables the technique to determine the thermal properties of low thermally conducting materials. The presence of the heat sink on the rare surface of the samples, in addition to its role as a mechanical support, it accelerates the heat conduction process through the sample to approach the steady-state condition in a relative short time. Thus, in this experimental set-up for low thermal conductivity samples with ≤ 2 W m−1 K−1 , the data evaluation procedures become more reliable and particularly simple. The heat through the source/sensor, in a form of a step-wise function, is supplied by the passage of an electrical current pulse. Meanwhile, the voltage (resistance/temperature) variations across the sensor were monitored using electrical bridge along with digital voltmeter (DVM) and a power supply. The data of electric current and the voltage across the sensor are measured via Wheatstone bridge arrangement and an interface card for data acquisition. A schematic drawing of the electric circuit is shown in Fig. 2. 2.2. The sensor The hot disk sensor is made of a nickel 10 m film covered from both sides with a thin insulating layer made of kapton. The kapton layer serves as an insulating layer that supports the heater (the film) in the hot disk as the depicted picture in Fig. 1. This arrangement fulfils the isothermal boundary conditions required by experimental setup. We are assuming that the heater has a negligible thickness and mass and is in perfect thermal contact with the sample. However, to investigate the possible deviation from this assumption an infrared (IR) camera is used to obtain IR images during the first stage (the very beginning) of the transient recording. The IR images in Fig. 3 display the average temperature variations across the sensor at two different timings in the very
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Fig. 2. Schematic drawing of the electrical circuit and apparatus used to perform the measurements.
beginning of the transient recording namely; at 160 ms as shown in Fig. 3(a) and at 330 ms in Fig. 3(b). For t = 160 ms, the Tavg variation (≈1 ◦ C) over the disk surface is attributed to disk manufacturing defects particularly due to the variation of the adhesive thickness between the copper tracks and the support. Note that the image in Fig. 3(a) displays two colder zones on the top and on the right side and arrow-heads like structures on the left side. Such characteristic behavior appears in the very beginning (t = 160 ms) but later tend to disappear while a warmer central cross-shaped structure is formed due to homogeneous heating distribution. This can be seen at later times (t = 330 ms) as shown in Fig. 3(b) where the variation of temperature difference yielded Td = Tmax − Tmin = 0.6 ◦ C. A full discussion of how to deal with this effect and others using the alternative optimal fitting procedures is presented later in Section 4.2.
3. The theoretical basis A full discussion of the theory has been presented in Refs. [2,9], however, the basis will be highlighted. According to Fig. 1, a sample of length 2l occupies the region −l < x < l, with the heater is placed in the plane x = 0. At the planes x = −l and x = l the sample is in contact with high conducting material (copper), here with thermal coefficients cu and acu . The temperature developments are governed by the solution to the following partial differential equations: ∂2 T 1 ∂T =0 − a ∂t ∂x2
0
∂2 TCu 1 ∂TCu − =0 aCu ∂t ∂x2
l
The initial and boundary conditions for one side are given by Tref = TCu = 0
0 < x < ∞ at
t≤0
∂T 1 = q x = 0 at t > 0 2 ∂x ∂TCu ∂T = x = l at t > 0 Cu ∂x ∂x TCu (t) = T (t) x = l at t > 0 −
TCu (t) → 0 x → ∞
at
t>0
where, q is the total output power per unit area dissipated by the heater. In order to establish the theoretical basis of the solution we will proceed starting with the ideal conditions as follows: (i) The heater has a negligible thickness and mass and is in perfect thermal contact with the sample. (ii) There is no thermal resistance between the sample and the heat sink. (iii) There are no heat losses from the lateral surfaces of the sample. Later we will discuss how close the actual experimental arrangement fits theses ideal conditions and how to detect and eliminate the errors due to the influence of these non-ideal conditions. According to Ref. [9], the temperature response function at x = 0, is given by: T (t) = where Fig. 3. (a) The average temperature variations Tavg = 1 ◦ C within the sensor after 160 ms. (b) The average temperature variations Tavg = 0.6 ◦ C within the sensor after 330 ms.
F(, t) =
ql √ F(, t)
t
(1)
√ ∞ n 1+2 ˇ ierf n=1
√ a
(2)
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Fig. 4. Schematic responses to temperature function versus time, tS represents the size of the optimal interval and is the characteristic time of the sample.
As it was mentioned above, q is the heat current density, and is the characteristic time of the sample which is defined in terms of the sample thickness and the thermal diffusivity as: =
l2 a
(3)
The coefficient ˇ is associated with the effect of the heat sink which is made of two cupper cylinders, in our case. ˇ ≈ −1 for perfect heat sink and ierfc is the error function integral [10]. Fig. 4 shows a typical temperature response function as a solution to the partial differential equations using the boundary and initial conditions that corresponds to the isothermal experimental arrangement. 4. Computational procedures 4.1. Standard fitting The principle of the computational procedures is based on fitting the theoretical temperature function to the experimental points. The standard fitting procedure (standard model) is based on a linear regression using least square fitting [8,9]. According to Eq. (1), the plot of experimental points Ti versus the calculated F(, ti ) should be a straight line if has its proper value. This equation predicts a zero intercept but real measurements showed a nonzero value T0 referred to an additional increase in the temperature due to contact resistance and design defects of the heater/sensor. The proper value of can be found using an iterative procedure to vary the characteristic time until the correlation coefficient calculated from Ti and F(, ti ) reaches its maximum. The slope of this straight line gives the thermal conductivity , and the proper iterated value is used in Eq. (3) to get the thermal diffusivity a. 4.2. Optimal fitting As mentioned in the previous section, in principle, there are two parameters whose values should be determined, namely; the thermal conductivity and thermal diffusivity a. However, due to the influence of the heater geometry, insulation layers (see picture in Fig. 1), and contact resistance, a third parameter To related to the baseline of the temperature response could be added to Eq. (2) so that the temperature response function becomes: a T =
t
√ ∞ n 1+2 ˇ ierf n=1
n √ a
+ T0
(4)
The optimal fitting model uses the computational procedures to select the optimum time interval for fitting the theoretical temperature function. It fits Eq. (4) to the points in the time interval (tB , tB + tS ), where tB and tS are representing the beginning and the size of the interval, respectively, as shown in Fig. 4 The procedures consider the beginning time as the varying variable of the selected interval within the transient recording. The results are plotted versus the beginning time and the optimal time interval is the interval within which the fitting is not sensitive to the interval size that cause a plateau in the plot. If tB is successively increased while tS is kept constant, a series of parameter values is obtained. If the time interval (tB , tB + tS ) is not suitable for determining a and , the results of fitting are unreliable and it will show considerable scatter with large deviations from the expected results. In order to verify these optimal fitting, we construct a mathematical model of the experiment as follows. In the first stage, the points were computed using Eq. (4) and simulating the measurement on a solid polymethylmetacrylate sample known as acrylic “Plexiglas”, the following values were used: l = 0.003 m, q = 1000 W m−1 , = 0.19 W m−1 K−1 , a = 0.12 × 10−6 m2 s−1 , T0 = 0.2 K, and ˇ = −0.954. The sampling rate was one reading per second, and the number of sampling points n = 300. In the second stage, noise was added by rounding the temperature coordinate of the points to seven digits. Then the points were re-computed using the smallest possible number of points (time interval). For example, if we have three unknown parameters in Eq. (4), we need at least three points for evaluation. In this situation – instead of the standard fitting procedures – we solve a system of three equations according to the following formula:
X − X0 X
RX =
(5)
0
where X0 is the simulated value used originally in the model and x is the value calculated using optimal fitting procedures. If the time interval is not suitable for estimation of parameters a and , the results are unreliable and relative differences are far from zero. To investigate the effect of the interval size within the model we have performed simulations using two different values of interval size: namely (a) tS = 10 s and (b) tS = 25 s, respectively. The results are depicted in Figs. 5 and 6. When we used a small window tS = 10 s, the curves are rather scattered and deviated from the original values of and a-presented by the straight horizontal line-used in the model, see Fig. 5. However, Fig. 6 shows the case of a larger window tS = 25 s, the fitted values of and a are nearly identical to the values used originally in the model up to tB = 50 s. It is very obvious from these figures that the results are influenced by the size of the window. It should be mentioned that the choice of 10 s and 25 s widows was basically to illustrate the effect of these fitting procedures and its possible potential use in other transient techniques. 5. Results and discussion The standard and optimal fitting procedures are evaluation models to seek an optimal time interval, in which the fitting procedures give results with minimum errors. The standard model is based on estimating the parameters using least-squares fitting when tB is constant while tS is successively increased and the results are plotted verses tS . In the standard analysis a number of points is used in the fitting procedure and can be defined as the interval [tB , tmax ]. Here tB corresponds to the number of points skipped at the beginning of the transient due to contact resistance and the insulation layers of the sensor and tmax corresponds to the maximum number of points during the transient recording. In the standard fitting procedures, there is no direct way of specifying the best interval. The only indicator
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Fig. 5. Results of the simulations using optimal analysis model. Thermal conductivity and diffusivity a as a function of the time window (tB , tB + tS ), for tS = 10 s.
Fig. 6. Results of the simulations using difference analysis modeling. Thermal conductivity and diffusivity a as a function of the time window (tB , tB + tS ), for tS = 25 s.
of good reliable measurements is using the so-called “difference graph”. It is a graph that displays the difference between the theoretical temperature increase (Tth ) as predicated by Eq. (1) and the experimental temperature increase (Texp ) from real measurements. Fig. 7(a) and (b) shows two different runs of reliable/stable and unreliable/disturbed measurements, respectively. The deviation from ideal case is clearly pounced in Fig. 7(b) either through the non-homogenous distribution of the data points or through comparing the relative magnitude of the differences between the maximum and minimum limits of the y-axis in both graphs.
In Fig. 7(b), the effect of the contact resistance and/or design defects of the sensor is also obvious at the initial stage of the transient recording. Measurements that have difference graph similar to the one depicted in Fig. 7(b) are usually disregards because it is a difficult task to specify the best interval (limited range of points) that can be used to get reliable results. On the contrary, in the optimal fitting model tB is the varying variable within the selected tS and the results plotted versus tB . As it was previously shown, the principle of the optimal analysis model is based on the assumptions that the time interval [tB , tB + tS ] is opti-
Fig. 7. The difference graphs of two different runs as an indicators of measurements stability and reliable results using the standard fitting procedures.
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Fig. 8. Values of the relative differences of the parameters a (x) and (+) vs. nondimensional time scale t/ (taken from Ref. [12]).
mal when the fitting is not sensitive to the interval size that cause a plateau in the plot. This can be seen by the optimal analysis of the real measurements of acrylic “Plexiglas” within the time window 0.1 ≤ t/ ≤ 1, depicted in Fig. 8. This figure shows the plot of the relative differences for both Ra and R , outside this region, Rx increases substantially and it deviates far from zero. These deviations are based on different fitting intervals shifted in steps up to the total time of the measurement. It is very obvious that these values are diminishing within the period [0.1, ]. In other words, this period will be the optimum time interval that will give reliable and accurate results. The distorting time in the very beginning (t/ < 0.1) of the transient event is related to the influence (defects) of the heat/sensor design as it was previously shown in the IR images. These defects are associated with the thickness and the mean thermal diffusivity of the layer/layers existing between the metallic heating pattern in the heater/sensor and the sample. It represents the deviation from the ideal conditions stated previously in Section 2. It should be mentioned that this is not only limited to the insulating layer supporting the heating element but also to any other layer between the heating element and the sample that contributes to the contact resistance. In principle, it is to all distortions that are related to the heater design, such as heat capacity of the metallic pattern in the sensor, the spacing between the successive strips in the pattern etc. Such distortions can be included and each will have its own
characteristic time that will affect the beginning of the transient recording. Fig. 8 is also showing that R values are very much scattered (deviates far from zero value) for all characteristic times greater than . This is attributed to rare side effects such as the thermal resistance between the rear surface of the sample and the heat sink which disturb the temperature development in the heater as soon as the heat pulse reaches the rear side of the sample. In the case of a real experiment, we should investigate, during the transient recording, all the effects which can cause the deviation of the experimental set-up from the ideal model and estimate their magnitude then reduce them accordingly. The first effect is due to the influence of the heater geometry. The theory assumes an ideal hot disk (i.e. a homogeneous hot plane of negligible thickness and mass that is in perfect thermal contact with the sample). The defects of the disk will cause the beginning of the measured temperature function to be distorted. This time interval, described by the characteristic time of the disc D , is not suitable for computing thermophysical parameters. Heat losses from the lateral sides of the sample present the second effect. We can eliminate them by optimizing the specimen thickness, according to Ref. [9], these losses are directly proportional to l2 and inversely proportional and the radius r of the sample. Thus, by proper choice of the geometry of the sample, the heat losses through the lateral sides of the sample can be reduced considerably just by keeping the relation l2 /r as small as possible. However, for relatively longer times when approaching the steady state condition, the heat losses increases, therefore long time interval also cannot be used. To investigate this effect we applied the model to the real measurements using the optimal size interval (tS = 25 s) and two different values of heating current namely; (a) I = 232 mA and (b) I = 663 mA. The corrsponding temperatures increase for these currents at the steady state condition were Ts = 1.9 and Ts = 8.0 C, respectively. Fig. 9 shows that the sensitivity of the measuring device was not sufficient at the heating current I = 323 mA. The results are widely scattered and roughly estimated values can be obtained only for tB = 10–15 s. However, the results in Fig. 10 are rather stable. The influence of the hot disk is clearly seen the diffusivity curves and the characteristic time of the disc can be estimated as D ≈ 5–10 s. In addition, the influence of heat losses from the lateral sides of the samples, for tB > 60 s, is very obvious in the plots. Hence, the time window within which the fitting procedure can be applied is from 10 to 60 s which corresponding to the interval within the range [0.1, ].
Fig. 9. Results of the optimal analysis of real measurements of acrylic “plexiglas”, thermal conductivity and diffusivity a as a function of the time window (tB , tB + tS ), for tS = 25 s, and I = 323 mA.
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Fig. 10. Results of the optimal analysis of real measurements of acrylic “plexiglas”, thermal conductivity and diffusivity a as a function of the time window (tB , tB + tS ), for tS = 25 s, and I = 663 mA.
Once the optimal time window has been determined, the desired thermophysical parameters and a can be calculated. Using this optimal time interval, we found the following: In the model, we obtained the values = 0.191 W m−1 K−1 and a = 0.120 × 10−6 m2 s−1 . In the real measurement of acrylic “Plexiglas” we obtained the values = 0.194 W m−1 K−1 and a = 0.122 × 10−6 m2 s−1 . These values correspond to variations of 1.5% in thermal conductivity and variations of 1.7% in thermal diffusivity. Our results are in a good agreement with the parameter estimation analysis (sensitivity coefficients) model [11–14] that yielded time interval within the range [0.07,] for the same material. It should be mentioned that these two models have different approaches. In the optimal model we use particular points with experimental or simulated noise while in the sensitivity coefficients model we directly use the temperature function response (no need for simulation) and the TPS-technique, has been used in Ref. [13]. 6. Conclusions The infrared camera images clearly displayed the deviations from the ideal conditions at the beginning of the transient recording due to design defects of a resistive sensor. The sensor known as the hot disk has been used as a plane heat source in the extended dynamic plane source (EDPS) technique. This technique is used for simultaneous measurements of the thermal conductivity, diffusivity and the specific heat and is based on a one-dimensional heat flow into a finite sample. By applying the optimal analysis model to the measured data from the temperature response (transient recording), we were able to deal with the deviations by finding the optimal time interval that yield more accurate and reliable results. The model is based on mathematical procedures that provide the selection of the optimal time interval within the total time of the transient recording. The obtained optimum time interval for acrylic “Plexiglas” sample was [0.1, ] which was in good agreement with the other selected time intervals obtained using other transient techniques and different evaluation model (sensitivity parameters estimations model). These fitting procedures can be used in similar techniques where the time interval plays a crucial part of the data analysis. It is an alternative to the standard fitting procedures which mainly based on linear regression and least square fit. More detailed analysis is planed in the near future to investigate the effect of using several time widows (intervals), say between 5 s and 45 s and the effect of power heat input on finding an optimal sample length. The
analysis will be on measurements in vacuum to reduce lateral heat losses from the surfaces.
Acknowledgments Great acknowledgements to Prof. Giovanni Maria Carlomagno and his research group from University of Naples in Italy for providing the IR images. The financial support from the University of Sharjah is also gratefully acknowledged.
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Biography Dr. Bashir Mohamed Suleiman, born 1955, has B.Sc. degree in nuclear engineering in 1979 from University of California at Santa Barbara, USA. He received a M.Sc. in solid state physics from University of Garyounis (Benghazi-Libya) in 1983 and Ph.D. degree in material physics from University of Gothenburg and Chalmers University of Technology (Sweden) in 1994. In 1998, he also received a “Docent” Diploma in Material Physics from University of Gothenburg, Sweden. He worked in Sweden as
scientific researcher (1989–1994) and then as an assistant professor (1994–1998) at the physics department Chalmers University Technology, Sweden. In November 1998, he joined the University of Sharjah, UAE. From 2000 until 2004, he was the head of the Applied Physics Department, University of Sharjah, Sharjah, UAE. Currently is an associate professor at the Applied Physics Department, University of Sharjah. Areas of interest are (1) Experimental investigations of thermal and electrical properties in solid. (2) Developing several transient techniques to measure thermal properties of solids. (3) Educational research in the role of applied physics within medical sciences, health sciences and engineering disciplines.