1 Introduction

The energy efficiency of buildings has become a priority mean of reducing a negative impact on the environment. Designing energy-efficient buildings requires a thoughtful choice of materials, which presumes an excellent knowledge of their thermal properties [1]. The values of these material properties can be determined using different methods which can be divided into two categories: steady-state and transient.

The principle of transient methods is to generate a heat flux inside the specimen (usually in the form of a step function) and to measure the temperature response (temperature dependence on time). Diffusivity a and conductivity λ are calculated by fitting the theoretical temperature function, using the least squares method, to the measured temperature response. This function is obtained by solving the heat equation under conditions that come close to the real experimental setup. The differences between the theoretical temperature function and the real experimental setup lead to systematic measurement errors. Thus, the more accurate the theoretical model is, the better the estimates of the thermophysical parameters are. However, this leads to a more complex analytical solution of the heat equation. In the extreme case, the analytical solution becomes impractical or unknown at all, and the problem must be solved numerically [2,3,4,5,6,7,8,9,10,11].

In the original extended dynamic plane source (EDPS) method the temperature of the heat source is determined by measuring its electrical resistance [12,13,14,15]. However, the measurement is affected by the heat losses from the specimen surface, which we eliminate by measuring the temperature in the middle of the heat source by a thermocouple. An important difference between the EDPS model and experiment is that the heat source capacity is not included in the theoretical model. Another difference arises from the theoretical model condition that the heat sink is infinitely large. Both effects can lead to unpredictable errors. This problem was solved in the new plane source (NPS) method by measuring the temperature of the heat sink with another thermocouple [9, 11]. Although the results of the method are excellent, it is rather impractical as the temperature must be measured in two places.

The numerical evaluation of the heat equation is carried out by finite elements method (FEM), which offers the possibility of including the geometrical and material structure of the experiment and computing the temperature response in a specific place [16,17,18,19,20]. However, the disadvantage of the method is that this response is obtained in numerical form and cannot be used for fitting the measured temperature response directly. In our paper [9], we developed 2D model for the transient plane source (TPS) method and showed how the numerical results of FEM simulation can be converted to analytical formula that was successfully fitted to the measured temperature responses.

The aim of this work is to verify the application of FEM for increasing the accuracy of the results obtained from EDPS measurements. This will be done with the data from numerical simulations of the experiment as well as from real measurements. The verification will also be performed with a new method derived as the modification of the EDPS method, for which the temperature function is unknown.

2 Theoretical Model

The theoretical model of the EDPS method is described in the paper [12] and is based on stepwise heating of the heat source and simultaneous measurement of its increasing temperature. The theoretical temperature function is an analytical solution of the model and has the form

$$T(t,a,\lambda ) = \frac{{qh}}{{\lambda \sqrt {\pi } }}D\left( {\frac{{\sqrt {{a\;t}} }}{h}} \right),$$
(1)

where the shape function [5] is

$$D(\xi ) = \xi \left( {1 + 2\sqrt {\pi } \sum_{n = 1}^\infty {\beta^n ierfc\left( {\frac{n}{\xi }} \right)} } \right)$$
(2)

and

$$\beta = \frac{{e - e_{HS} }}{{e + e_{HS} }}$$
(3)

Here, a and λ are the thermal diffusivity and the thermal conductivity of the specimen. \(e = \lambda /\sqrt {a}\) is its effusivity, eHS is the effusivity of the heat sink (Cu), q is the heat current density at the heat source, and h is the thickness of the specimen.

3 Experiment

The experimental arrangement of EDPS method is shown in Fig. 1a. It should be emphasized that the configuration is cylindrical and symmetrical so that two identical pieces of specimen are required. The heat source is realized as a thin nickel disk in the form of double spiral with Kapton layers on both sides. Its temperature is measured using the thermocouple made from chromel and alumel wires with a diameter of 25 μm. The wires are electrically connected via an aluminum circle with a thickness of 10 μm and a diameter of 6 mm. The circle removes the inhomogeneity in the temperature field caused by the striped structure of the heat source. The diameter of the test specimen is 61 mm which consists of cylinders with a height of 2.95 mm for glass and 2.84 mm for PMMA. The entire assembly, bonded by silicone oil, is firmly screwed between two large copper blocks with a diameter of 60 mm and a height of 49 mm, which serve as the heat sink. Once the experimental temperature of 25.0 °C has been stabilized, the current in the heat source is switched on and at the same time the voltage on the thermocouple starts to be sampled with a nanovoltmeter.

Fig. 1
figure 1

The experimental setup of the EDPS method (a) and the EDPS1 method (b)

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A modification of the EDPS method, designated as EDPS1, consists in replacing one piece of the specimen with a material whose thermophysical parameters are known. The advantage of this method is that only one piece of the specimen is required. The disadvantage is that the analytical temperature function is unknown.

4 Simulation of the Experiment

The analytical model of EDPS method considers a heat source capacity of zero and an infinitely long heat sink, which does not correspond to the real experiments. This means that the specimen parameters values determined by formula (1) deviate slightly from the actual values. This deficiency can be removed by computing the temperature function numerically using a FEM simulation based on the real parameters of the experiment.

The numerical model considers the actual geometry and material properties of the heat source so the real value of the heat source capacity is included in the simulation. The contact thermal resistance between heat source and specimen can reliably be neglected because the surfaces are planar, smooth, and covered with silicone oil, which is used to reduce the contact thermal resistance. Thus, the thin gap of the air is replaced by the oil, which has a much higher thermal conductivity. This applies to all specimen materials. Moreover, the parts are firmly screwed together so the gaps between Kapton and Al foil and also between Al foil and specimen are considerably thinner than the Kapton foil itself. The oil film is not included in the FEM model because its thickness cannot be measured. In the FEM model, the heat sink is finite and its height corresponds to that of the real heat sink.

The FEM model of the EDPS method is one-dimensional because the heat flow in the central part of the specimen in the radial direction is negligible. Therefore, the specimen radius in the model can be reduced to 0.1 mm to speed up the calculation. Only one half of the experimental setup is modeled, because of its mirror symmetry with respect to the heat source.

The Gmsh software is used to create the geometric model of the FEM simulation [21]. The mesh structure and size at the heat source are shown in Fig. 2. Here at the interfaces, the mesh is much denser than in the relatively large heat sink. The runtime of the simulation was 60 s, but the maximum time in the computed temperature response was 100 s.

Fig. 2
figure 2

The central part of the mesh created by Gmsh in EDPS simulation

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The thermocouple wires and the aluminum circle are represented in the model by only 20 μm aluminum foil. The FEM simulation is performed using the Elmer FEM simulation software [22] with the coordinate system = Axisymmetric, the adaptive time stepping method to a relative error = 0.001 and the total power 1.0 W abruptly switched on at time t = 0. All surface areas of the assembly are considered in the model as zero flux boundaries. The model utilizes the table values of material parameters of the real experimental setup presented in Table 1. The specimen parameters are set differently in each simulation.

Table 1 Material parameters used for the FEM simulations [23,24,25,26,27]
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In EDPS1 modification, the mirror symmetry is removed and both sides of the heat source are modeled as shown in Fig. 3. The large sensitivity coefficient of the method means that a small change in the value of the thermophysical parameter will cause a large change in the measured temperature response, which is only possible if the heat flux to the specimen is large enough. For the method to be sensitive, the heat flux to the specimen side should be higher than to the known material. It can easily be derived from Eqs. (1) and (2) that the ratio of heat fluxes is equal to the ratio of effusivities. Therefore, the specimen should have a higher value of effusivity than the known material.

Fig. 3
figure 3

The central part of the mesh created by Gmsh in EDPS1 simulation

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5 Experiment Evaluation

The EDPS experiment evaluation means determining the values of the unknown specimen parameters from the measured temperature response, which we will refer to as “input response”. This can be done using following steps:

  1. (1)

    The initial guesses a1 and λ1 of the unknown parameters are determined by fitting the analytical temperature function given by Eq. (1) to the input response.

  2. (2)

    A new temperature response [ti, Ti] is computed by FEM simulation using these initial guesses of the parameters and other geometrical and material parameters of the experiment.

  3. (3)

    The improved analytical formula of the temperature function is very similar to Eq. (1) and has the form

    $$T_1 (t) = \frac{qh}{{\lambda \sqrt {\pi } }}E\left( {\frac{{\sqrt {at} }}{h}} \right)$$
    (4)

    where the shape function E(x) is unknown but can be determined in the form of points [xi, E(xi)] as follows:

    $$x_i = \frac{{\sqrt {a_1 t_i } }}{h},\;E(x_i ) = \frac{{T_i \lambda_i \sqrt {\pi } }}{qh}$$
    (5)
  4. (4)

    Once we have the shape function E(x) in a numerical form, we can convert it to an analytical form using spline interpolation. The quantities q and h as well as the initial guesses a1 and λ1 used in (5) must be the same as in FEM simulation in step 2.

  5. (5)

    Finally, Eq. (4) and the shape function E(x) in an analytical form are fitted to the input response yielding new parameter estimates a2, λ2 that are better than the initial guesses a1, λ1.

This algorithm can be verified by simulating the experiment, i.e., by computing the input response by using FEM, the real geometrical and material structure of the experiment, and values a0, λ0 of the specimen parameters determined from the previous measurements. These values are considered to be true as they exactly match the input response as far as the FEM model is accurate. We can then evaluate the simulated input response via steps 1 to 5. The advantage of simulating the experiment is that the true values of parameters a0, λ0 can be compared with the initial guesses a1, λ1 and the final parameter estimates a2, λ2.

In the case of the EDPS1 experiment evaluation, where the analytical temperature function is not available, the initial guesses must be computed by the EDPS method and are therefore highly erroneous. Thus, the described procedure must be used recurrently, where the final parameter values are used as initial guesses in the next cycle, and in each cycle, only one FEM calculation in step 2 is needed (Fig. 4).

Fig. 4
figure 4

The schematic showing the steps and the procedure used to calculate improved estimates of the thermal conductivity and diffusivity

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6 Results and Discussion

Figure 5 illustrates the time dependence of the heat source temperature in EDPS experiment determined by analytical formula (1) and the FEM simulation. The specimen is glass with parameters values of a0 = 0.5 mm2.s−1 and λ0 = 1.0 W.m−1.K−1, the specimen thickness is 3 mm, and heating power is 1 W.

Fig. 5
figure 5

The temperature response of EDPS method computed using analytical formula (1) T1 and using FEM simulation T2

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Table 2 shows the results of the evaluation of the EDPS measurement simulation. Firstly, the input responses are calculated by FEM using the rounded values of real thermophysical parameters. These values are considered in the simulation as the true values a0 = 0.5 mm2.s−1 and λ0 = 1.0 W.m−1.K−1 for glass and a0 = 0.1 mm2.s−1 and λ0 = 0.2 W.m−1.K−1 for PMMA. The input responses are then evaluated in steps 1 to 5 as described in Sect. 5. As seen in Table 2, the new estimates are significantly better than initial guesses.

Table 2 Results of evaluation of the EDPS measurement simulation
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Table 3 shows the results of the evaluation of the EDPS1 measurement simulation of glass, where the known material is PMMA with a thickness of 50 mm. The input responses are calculated by FEM using the rounded values given above. Since there is no analytical formula for the EDPS1 method, the input responses are evaluated using the EDPS method and the initial guesses a1 and λ1 are therefore very rough. Thus, the evaluations must be done 3 times according to steps 2 to 5 in Sect. 5 until the estimates reach relative errors of about 1%. The results of one evaluation are used as the initial guesses for the next evaluation. As seen in Table 3, each time the estimates are better than the previous ones.

Table 3 Results of evaluation of the EDPS1 measurement simulation
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Table 4 shows the results of the real measurement of the glass specimen by the EDPS1 method using the known PMMA material with a thickness of 2 × 2.84 mm. The thermophysical parameters are known from the measurements with other methods as a0 = 0.507 mm2.s−1 and λ0 = 1.017 W m−1 K−1 for glass and a0 = 0.1143 mm2 s−1 and λ0 = 0.1915 W m−1 K−1 for PMMA [9,10,11]. The measured temperature responses are evaluated by the analytical formulas of the EDPS method which leads to very rough initial guesses a1, λ1. Therefore, the evaluation according to steps 2–5 in Sect. 5 has to be done 3 times until the estimates reach relative errors of about 1%. It should be emphasized that while the thermophysical parameters of PMMA are known and used in all FEM simulations, the glass parameters are considered as unknown and used only in comparison with the final estimates in Table 4.

Table 4 Results of the measurement of glass specimen using EDPS1 method
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7 Conclusions

The conventional analytical formula for EDPS method used for the evaluation of experiments has limited accuracy because it does not consider the heat source capacity and heat sink dimensions. This work eliminates these shortcomings by combining the analytical formula with the FEM numerical simulation of the experiment. However, FEM provides the temperature response in the numerical form, which cannot be used in fitting measured response. Therefore, a method of transformation of a function in numerical form into an analytical one is introduced and applied. To confirm this combined approach two FEM simulations of the experiment and one real experiment are conducted.

In the case of EDPS simulation of the experiment, the initial guesses of the thermophysical parameters are determined by the analytical formula with relative errors ranging from 1 to 5%, as seen in Table 2. Since the values are quite close, only one FEM run is required to reduce the relative errors to be below 0.5%. In the case of the EDPS1 simulation of the experiment, the initial guesses are computed using an analytical model for the EDPS method. The relative errors of up to 38% are reduced to less than 1.4% by 3 repeated FEM runs. In the case of the EDPS1 experiment, the initial guesses with relative errors up to 37% are reduced to less than 1.5% by 3 FEM runs. The improvement caused by FEM can be seen in Tables 2, 3, and 4, which show the results of the analytical and numerical FEM evaluations. These results were compared with reliable values in the form of relative differences.

In this work, the uncertainty of the thermal conductivity and diffusivity measurement is not directly stated. The accuracy of the results is assessed only by comparing with the reliable values of the specimen parameters determined using other methods for which the uncertainty was stated to be less than 1% [9,10,11]. Therefore, the specimen effusivity can be calculated with similar accuracy. If the specimen thickness is large enough, the specimen effusivity can be measured using EDPS setup directly with good precision.

We can conclude that the new method of transient measurements evaluation using FEM has been successfully verified on both simulations of the experiment and in real measurements. It has been shown that in the case of very rough initial guesses, accurate results were reached by repeating the numerical evaluation. However, this situation could be avoided using better initial guesses obtained for example from literature or another method, even if they are not very accurate. Finally, it should be emphasized that the used numerical model was not absolutely accurate. The real shape of the heat source was spiral, but the model considered a full disc. The real thermocouple consisted of two wires and a small aluminum circle, but the model used only the circle. Thus, a small inhomogeneity in the temperature field, caused by the striped structure of the heat source, was smoothed by the aluminum circle.

Other effects were the heat losses from the heat sink surface, which were not included in the numerical model. The thermal resistance between the specimen and heat sink was also ignored. Despite these discrepancies, the results of real measurements show a relative error of about 1%.

The most important contribution of this paper is the method for transforming the temperature function from the numerical to the analytical form, which can be used for fitting measured data. It can be used wherever the analytical solution of the differential equation of the problem is known but not accurate enough. The advantage of this method is that it computes much faster than the full numerical fit. The disadvantage is that it has not been mathematically proven yet, but it works perfectly as verified in this paper.