Influence of heat loss through probe electrical leads on thermal conductivity measurement with TPS method

WANG Yu-wei， LI Yan-ning

(National Key Laboratory of Precision Testing Techniques and Instrument, Tianjin University, Tianjin 300072, China)

0 Introduction

Insulation materials play an important role in building energy conservation, aerospace insulation and automotive industry[1-3]. Thermal conductivity is a key parameter to evaluate the thermal insulation properties of insulation materials. Therefore, the accurate measurement of the thermal conductivity of insulation materials is of great significance.

The methods for measuring the thermal conductivity of materials are broadly classified as steady state methods and transient methods[4]. As to the steady state methods, the guarded hot plate (GHP)[5]method is regarded as the most commonly used method for measuring the thermal conductivity of insulation materials. The principle of this measurement is to establish a steady temperature gradient through a certain heat flux in the longitudinal one-dimensional direction of the sample. Then the thermal conductivity is calculated according to the Fourier law. Nevertheless, this technique requires a long time to establish a steady temperature gradient, and requires a relatively large sample size. The basic principle of the transient method is to heat the testing material in a short time. The thermal conductivity of the material is calculated by recording the surface temperature change of the probe with time. Compared with steady state methods, the transient methods have the advantages of short measurement time, a wide range of samples and high adaptability. At present, transient hot wire method[6]and transient plane source (TPS) method are the main applications of transient methods[7].

The TPS method is a transient method firstly developed by Gustafsson in 1991[8], which can obtain thermal conductivity, thermal diffusivity and volume specific heat capacity of the material simultaneously via a single transient measurement. Drawing on the ideal model of the TPS method, the heating power is assumed to be constant during the measurement. However, in fact, the heating power is influenced by the heat loss through the electrical leads. Therefore, the accuracy of thermal conductivity measurement will be reduced. Based on the ideal model of the TPS technique, this paper presents a model to correct the heat loss through the electrical leads so as to solve the foregoing problem. A series of experiments with different materials have been conducted within the proposed model. The results are presented and discussed in this paper.

1 Measurement principle of TPS method

The principle for measuring the thermal conductivity of material by the TPS method is based on the transient temperature response of the plane heat source in the infinite medium which is subjected to an abrupt electrical pulse[9]. The core element of this method is the TPS probe, which is composed of a bifilar spiral structure with an etching of metal foil, as seen in Fig.1. The metal foil is covered on both sides by insulation film (Kapton or Mica) for protection and electrical insulation.

During the experiment, the probe is tightly placed between two pieces of samples so that forming a sandwich structure as shown in Fig.2. The probe not only acts as a heating element, but also as a temperature sensor for recording the temperature of the probe[10]. When a constant heating power is applied to the probe, due to the current heat effect, the probe temperature increases and the heat from the probe is transferred to the sample on both sides through conduction. By recording the changes of its voltage, the relations between the probe temperature and time can be obtained, which reflect the thermal properties of the testing material. In doing so, the thermal conductivity of the testing material can be calculated. In order to facilitate the theoretical analysis, the bifilar spiral structure is simplified into an equidistant concentric ring structure in the theoretical calculation.

Fig.2 Schematic diagram of measurement structure

The probe’s temperature rises when electric current pass through the probe. The temperature increase at any point on the probe plane at timetis expressed as[11]

(1)

whereP0is the heating power applied to the probe;λis the thermal conductivity of the testing sample;ais the radius of the outermost ring of the probe;mis the number of concentric rings of the probe;I0(x) is the first class modified Bessel function of zero-order;τis the non-dimensional time parameter defined as

(2)

θ=a2/κ,

(3)

whereκis the thermal diffusivity of the testing sample;θis the characteristic time described by Eq.(3).

The total length of the metallic wire is

(4)

Then the average temperature increase of the probe surface can be obtained by averaging over the length of the concentric rings

(5)

whereD(τ) is a dimensionless time function given by

(6)

Eq.(5) is the ideal model for the average temperature increase of the probe surface by TPS method.

During the measurement, the temperature increase of the probe surface is obtained by measuring the change of the probe resistance. The relationship between the resistance increase in probe and time is

(7)

From Eq.(7), the temperature increase of the probe surface can be obtained by

(8)

whereR(t) is the resistance of probe at timet;R0is the initial resistance of probe;αis the temperature coefficient of probe resistivity.

2 Correction for heat loss through electrical leads

In light of the ideal model of the TPS method, the heating power is considered to be constant. However, there is a heat transfer via the electrical leads during the measurement that will cause power loss. Consequently, the actual heating power of the sample is less than the rated input power of the probe, which reduces the accuracy of the thermal conductivity measurement. In order to improve the accuracy of this measurement, the heat loss caused by the heat transfer via the probe leads is analyzed and corrected. A corrected model of the average temperature increase of the probe surface is proposed.

The etched-out leads of the probe are of the shape depicted in Fig.3.

Fig.3 Probe dimension diagram

With the assumptions that the lead pattern is wide enough to avoid self-heating and the ambient temperature remains constant during the measurement, the loss of power through the electrical leads ΔPlis given by

(9)

where ΔTlis the average temperature difference between the edge of the sensor and the points of contact with the heavy electrical leads in heating time;Rlis the heat transfer thermal resistance of the electrical leads.

(10)

(11)

When heat transfer occurs on the leads, the heat transfer thermal resistance of the leads is

(12)

According to Eq.(9), the heat loss through the leads can be written as

(13)

wheredmis the thickness of the metal used as sensing material of the probe. The influence of the insulation layers is ignored because the heat loss via the contacts to the metal pattern is normally higher than the conduction via the insulation layers;λmis the thermal conductivity of the metal material;H,h,L,lare given in Fig.3;γis a correction factor, which is determined by using a standard reference material with low thermal conductivity. The actual heating power can be written as

(14)

By replacingP0in Eq.(5) withPin Eq.(14), the average temperature increase of the TPS probe can be rewritten as

(15)

The above formula is the corrected probe temperature increase model after considering the leads heat loss. In the following, the influence of the heat loss through the electrical leads on the measurement of the thermal conductivity is studied experimentally based respectively on Eqs.(5), (13) and (15).

3 Experiment

3.1 Experimental materials

In order to evaluate the influence of the heat loss through the electrical leads on the thermal conductivity of different materials, the following eight materials were examined in this study, namely black rubber sheet, polymethyl methacrylate (PMMA), wood, marble, stainless steel, lead, industrial pure iron and brass. Three kinds of insulation materials, namely black rubber sheet, extruded polystyrene (XPS) and polyurethane, were selected for comparison experiments to compare the heat loss before and after correction. The standard values of thermal conductivity of these materials are polyurethane of 0.022 W/(m·K), XPS of 0.032 W/(m·K), black rubber sheet of 0.037 W/(m·K), PMMA of 0.2 W/(m·K), wood of 0.445 W/(m·K), marble of 1.822 W/(m·K), stainless steel of 14.5 W/(m·K), lead of 34.8 W/(m·K), industrial pure iron of 74.4 W/(m·K) and brass of 119.1 W/(m·K).

3.2 Experimental parameters

When performing the measurements, the TPS probe was sandwiched between two identical samples, as shown in Fig.2. The probe simultaneously acted as the heat source and the temperature sensor. In order to ensure that the thermal effect of the probe was the only factor that caused the temperature increase of the sample, before the experiment, the experimental device was placed in a room with constant temperature so as to keep the experimental device consistent with ambient temperature.

In the TPS method, it is assumed that the probe is placed in a sample that is infinitely large. Therefore, the heat flow can’t reach the sample’s boundary. The distance travelled by the heat flow during the measurement is defined as the probing depthDby

(16)

which depends on the sample’s thermal diffusivityκand the measuring timet.

So the minimum size of the sample must meet

W>2D+2a,

(17)

H>D,

(18)

whereWis the minimum width of the sample andHis the minimum thickness of the sample.

In addition, in order to obtain a stable thermal conductivity and thermal diffusivity, according to the sensitivity coefficient theory, the measurement time should be between 1/3 of the characteristic time and the entire characteristic time[12],

(19)

The size of the sample in this paper meets the above requirements. One probe with a radius of 6.4 mm was used to measure PMMA, wood, marble and stainless steel. The rest of materials were measured by another probe with a radius of 14.6 mm. The relations of Eqs.(16)-(19) are used together to select a suitable heating time and heating power[13].

4 Results and analysis

Firstly, the eight different materials were measured by the hot disk thermal constant analyzer. The temperature response data of the probe with different materials were obtained by the data acquisition system. Relying on the obtained temperature response data, the heat loss through the leads ΔPlunder different materials was calculated according to Eq.(13). The experimental results are shown in Table 1.

Table 1 Heat loss through leads under different materials

The proportion of heat loss in heating power ΔPl/P0under different materials was calculated from Table 1. The results are shown in Fig.4.

Fig.4 Influence of heat loss through electrical leads on measurement under different materials

In line with Table 1 and Fig.4, it can be found that the influence of the heat loss through the electrical leads on the thermal conductivity measurement (namely ΔPl/P0) is different in the aforementioned eight materials. The influence of the heat loss increases when the thermal conductivity of the material decreases. The major reason of this finding is that the larger the thermal conductivity of the sample is, the faster the heat transfer rate is, the higher the heating power is required, and the corresponding temperature difference (the temperature difference between the hot disk and the leads) is relatively small. Eq.(9) shows that the heat loss is essentially proportional to the temperature difference between the hot disk and the leads, so ΔPl/P0is smaller. Therefore the following conclusions can be drawn: the influence of leads heat loss on the measurement increases with the decrease of the thermal conductivity of the material. When the thermal conductivity of the material is greater than 0.2 W/(m·K), the influence of leads heat loss on the measurement is less than 0.16%, which can be ignored. As for the materials with low thermal conductivity, the heat loss through the leads must be corrected.

Based on the above analysis, the thermal conductivities of black rubber sheet, XPS and polyurethane were measured respectively by the ideal probe temperature increase model and the corrected probe temperature increase model. The results of thermal conductivity values are listed in Table 2. Furthermore, the relative errors of thermal conductivity of black rubber sheet under different heating power from 0.07 W to 0.14 W with 0.01 W interval were studied with different models and different probes.

Table 2 Thermal conductivity measuring results with or without leads heat loss correction

Columns 1, 3 and 5 of Table 2 indicate that the thermal conductivity values of the three kinds of insulation materials obtained by corrected model are less than that produced by ideal model. The column 6 shows that after the correction of the leads heat loss with Eq.(15), the relative errors are significantly reduced compared with the ideal model by 1.04%, 1.20%, 1.59% for the three materials respectively. This implies that, on the one hand, the leads heat loss does have a great impact on the measurement of low thermal conductivity materials, and on the other hand, the proposed corrected model can effectively improve the measurement accuracy. The foregoing results are consistent with the previous conclusion that the influence of leads heat loss on the measurement increases with the decrease of thermal conductivity.

In the light of Fig.5, regarding the same probe, the relative errors of the thermal conductivity of the black rubber sheet after correction are reduced. With regard to different probes, the relative errors of the thermal conductivity calculated by the ideal model with small size probe are greater than that with large size probe. The relative errors of the thermal conductivity calculated by the corrected model with different size probes are basically the same. This means that the proposed corrected model is suitable for probes with different sizes. For both probes, either before or after the correction for heat loss, the relative errors of thermal conductivity of black rubber sheet measured at different heating power vary little. Hence, the influence of the heat loss on the measurement of thermal conductivity is basically unchanged under different heating power.

5 Conclusion

In this work, the heat loss through the electrical leads, which affects the measurement accuracy of thermal conductivity, has been studied. A model for correcting the heat loss is proposed. Based on a series of experiments, the following conclusions are drawn:

1) The influence of the heat loss on the measurement accuracy increases when the thermal conductivity of the material decreases, and the influence on materials with thermal conductivity greater than 0.2 W/(m·K) can be ignored.

2) When measuring the low thermal conductivity materials, it is necessary to correct the heat loss through the electrical leads, which can effectively improve the measurement accuracy. In addition, it is best to use a larger size probe so as to reduce the proportion of heat loss in the measurement, and thereby reducing the influence of heat loss through the probe leads on the measurement.

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