Evaluation of transient measurements of thermophysical parameters using Finite Elements Method.

The transient plane source (TPS) method was used to measure the thermal diffusivity and conductivity of silica glass and polymethyl methacrylate (PMMA). The results were evaluated by an analytical model and also by a numerical model created by the finite elements method (FEM). The results of both models and both materials were compared, and the difference was less than 0.4%.


Introduction
Transient methods lie in the dynamic generation of the heat flow and the measurement of the temperature response.Thermophysical parameters of the specimen are determined by fitting the theoretical temperature function to the temperature response.
In the TPS (transient plane source) method the temperature of the heat source is found out from the measurement of its resistance.The method is characterized by the cylindrical symmetry of the heat flow in the specimen.The theoretical model assumes that the sensor is completely surrounded by an infinite specimen.In the experiment the specimen is finite, as seen in figure 1, so only the part of the temperature response until the heat flow reaches the specimen surface is evaluated.
The theoretical temperature function was published in [1-5] for a specific number of circular line sources in spite of the fact that the actual heat source consists of a bifilar spirals.So the model of concentric circular strips proposed in [6] describes experiment better, despite the model does not contain material of the strips and kapton insulation.The aim of this work is to verify the analytical model by finite elements method (FEM) including the geometrical and material parameters of the heat source.

Analytical model
The TPS experiment set up is shown in figure 1 and the theoretical temperature is given by ( ) ( ) where the shape function is IOP Publishing doi:10.1088/1742-6596/2628/1/012020 2 R is outer radius of the sensor, P is the input heat power and N is the number of strips.The shape function (2) for A = 1/4 corresponds to the strip model of the heat source [6].In the analytical model of TPS method the temperature resistance caused by the kapton insulation of the heat source brings about a temperature difference between the heat source and the specimen surface.This temperature difference can be considered constant because the insulation layer is thin and consequently the relaxation time is less than 10 ms.To reduce its influence, the temperature offset must be used as a third fitting parameter.So in practise the following temperature function is used in parameter fitting where w is the variable which suppresses the effect of the contact resistance between heat source and specimen.

Numerical model
Numerical calculations are performed by the finite element method (FEM) applying the Elmer software [8].The geometry of the model and its numerical network are created by the GMSH software [7].
The FEM model has cylindrical and mirror symmetry, and its mesh is shown in figure 2. The heat source is modelled as 16 concentric circular strips with a width of 200 µm and a thickness of 10 µm.Between the strips and the specimen is placed the kapton insulation with the thickness of 10 µm.The heat in the strips was suddenly turned on at time t = 0.The adaptive time stepping was set to a relative error of 510 -4 and a minimum time step size Δ t = 10 -5 s.This small value is necessary in the initial times of sudden switching of the heating current.The total time of the simulation was set to 60 s.Table 1 presents the standard values of material parameters used in the simulation.The boundary conditions set in the simulations had no effect on results, because in the time of the measurement (t = 60 s) the heat flow did not reach the boundaries of the specimen as illustrated in figure 3.In the FEM simulation three modifications of the heat source structure were considered.(I) The material of the heat source is the same as the specimen.(II) The material of the heat source strips heat source specimen z is specified as nickel.(III) The material of the strips is nickel and the material of the other parts of the heat source is Kapton.Various structures of the heat source influence its heat capacity and the thermal resistance between the specimen and the nickel strips, which temperature is measured.So the temperature responses in the FEM simulations are slightly different.Similar simulations were performed with the PMMA specimen..

Figure 2.
The geometry of the FEM model for the TPS method and its numerical mesh.A zoom illustrates the strips and kapton insulation of the heat source.

Experiment
In the TPS method, the heat flow was generated by a double nickel spiral with Kapton coating.The temperature response was determined by measuring the resistance of the heat source using the bridge [3].At first the experimental temperature was stabilized to 25.0 C.Than the measurement started by sampling the voltage with a nanovoltmeter.When the first sample measuring has finished, the VMC signal (voltmeter complete) switched on the heating current in the heat source.So that the difference between middle of the first sample and switching time was only about 50 ms.The number of samples was always 300.
The specimens consist of cylinders with a diameter of 61 mm and with a height of 2.95 mm in case of glass and 2.84 mm in case of PMMA.All cylinders and the heat sink were covered by silicone oil and firmly screwed together to improve the thermal contacts.

Results and discussion
The experiment evaluation means determining the values of the unknown parameters of specimen from the measured response.The first step of the evaluation is fitting the analytical solution of the model ( 4), to this response and get the initial guesses a1 and 1 of the unknown parameters.The second step consists in computing the FEM response using these initial guesses of the parameters.This response is in the numerical form as points [ti, Ti], but we need the analytical one, which can be used in fitting measured response.
This issue is solved by connecting the numerical solution with analytical.We will suppose that the difference between analytical formula (4) and real response is small.Then we can expect that the analytical formula of the solution, which we have in a numerical form [ti, Ti], will be very similar to equation (1) and will have the form where the shape function E(x) is unknown, however we have a lot of points   When we have the shape function in a numerical form, we can transform it to an analytical form using spline interpolation.Quantities R and P used in (5) must be the same as in FEM calculation of the response [ti, Ti].Parameter values a1 and 1 are the initial guesses and also must be equal to those used in FEM calculation.As the small shift in temperature can occur in the measured data as well as in FEM simulation, the experiments proved that the better result can be obtain using the additional variable w.Finally, the equation ( 7) and the shape function E(x) in an analytical form are fitted to the measured temperature response giving new better parameter estimates.

(
) Table 2.The results of fitting thermal diffusivity and conductivity using three shape functions determined by FEM and one analytical shape function. represents the difference between analytical and numerical models.2 shows the results of the evaluation of both thermophysical parameter in glass and PMMA by the three shape functions obtained by FEM simulation.In (I) the material of the heat source is the same as that of the specimen.In (II) material of the heat source is nickel and in (III) the heat source is covered by kapton.There are also the results gained using the analytical shape function (4).The evaluations were carried out using one measured file in case of glass and also one file in case of PMMA.The superb consistency of the results in table 2 is the proof that fitting three parameter a,  and w are sufficient.So the variable shift in temperature eliminated the influence of the heat source capacity satisfactory and the use of the variable time correction introduced in [1] would be redundant and only increases the uncertainty of the measurement results.However, this could only be applied if the time difference between the first sample and the switching on of the heating current is very small, as described in part 4.
Conclusions TPS method was used to thermal diffusivity and conductivity measurement of silicate glass and PMMA.One temperature response measured on each material was evaluated by one analytical model and three numerical models based on FEM.The results obtained by various models were in a very good agreement.The differences were less than 0.4% for both materials.
The method of determining the shape function by FEM can be used in the evaluation of the transient methods of thermophysical parameter measurement, where the analytical temperature function is not known.

Figure 1 .
Figure 1.The set up of the TPS method.

Figure 3 .
Figure 3.The temperature field in the specimen at the end of the simulation.

Table 1 .
Material parameters used for the FEM simulations