1. Introduction

2. Literature review

Figure 1. Thermal conductivity of mineral wool as a function of temperature.

Figure 1. Thermal conductivity of mineral wool as a function of temperature.

Table 1. Thermal properties of mineral wool insulation with density more than 26 kg/m³ [15].

3. Description of the test method

3.2. Experimental setup

The experimental study involves a cuboid specimen composed of a tested material that is subjected to the heating in the Carbolite Gero CWF 13/36 laboratory-grade chamber furnace. This furnace is specifically designed for high-temperature applications, with a maximum operating temperature of 1300°C and a heating chamber measuring 320 mm × 250 mm × 450 mm, providing a total volume of 36 liters. The furnace is equipped with silicon carbide heating elements, known for their superior thermal stability and temperature uniformity. The furnace temperature is controlled by a CC-T1 touch screen temperature controller, offering precise control over 24 segments that may be set as ramp, step, or dwell and configured to control relays. The furnace’s safety over-temperature controller provides protection against overheating.

To minimize lateral heat flow during the heat process, the cuboid specimen being tested is surrounded by insulation material on the front, left, and right sides. On the back surface, an aerated concrete brick adheres to the sample, which allows for precise fixation of thermocouples (see Figures 2 and 3). The sample is exposed from the top, so heat flow is expected from top to bottom. The tests were carried out for samples of 100 mm × 100 mm × 100 mm. Seven type K thermocouples (denoted asT1, T2, T3, T4, T5, T6, T7) were placed inside the specimen at known vertical intervals as shown in Figure 2 (Detail 1) and Figure 3a. PicoLog 6 software in conjunction with the TC-08 Pico Data Acquisition (DAQ) devices offers a comprehensive solution for capturing, recording, analysing, and visualising data from thermocouples placed inside the tested specimens. This allowed for effortless monitoring and recording of temperature data from all thermocouples at 1-second intervals. Once the data is captured, PicoLog 6 software enables the secure storage of the acquired temperature measurements. This ensures that valuable data is preserved for analysis.

Figure 2. Sketch of a specimen inside the furnace.

Figure 2. Sketch of a specimen inside the furnace.

Due to the small dimensions of the sample, the thermocouples were horizontally shifted relative to each other. However, it is worth noting that for 1-D heat transfer, isotherms in the specimen are horizontal, hence only the vertical coordinate of the measurement point is relevant. The horizontal offset of the thermocouples allows their vertical distances to be smaller. The true locations of thermocouples are precisely verified after each test to provide the reliable data for the calculation.

Figure 3. The test specimen inside the electrical furnace: (a) fixing position of a thermocouples by the use of an aerated concrete brick at the back of the furnace; (b) sticking a mineral wool specimen on the thermocouples; (c) and (d) insulating the specimen from all-around.

Figure 3. The test specimen inside the electrical furnace: (a) fixing position of a thermocouples by the use of an aerated concrete brick at the back of the furnace; (b) sticking a mineral wool specimen on the thermocouples; (c) and (d) insulating the specimen from all-around.

3.3. Calculation method

This section aims to present the methodology employed for calculating the diffusivity and provide the calculation process for a specific time point. Using Eq. (3), the function of the thermal diffusivity $\alpha \left(\theta \right)$ can be obtained knowing the 2^nd order derivative of the temperature with respect to distance coordinate and the derivative of the temperature with respect to time:

$$\alpha \left(\theta \right)=\left(\frac{\partial \theta }{\partial t}\right)·{\left(\frac{{\partial }^2\theta }{\partial x^2}\right)}^{-1}.$$

(5)

Experimental data provide spatial and temporal distribution of temperature in a material at distinct positions and specified time steps. It allows the approximation of the derivatives in Eq. (5) based on the experimental results and then the calculation of the thermal diffusivity.

To calculate the thermal diffusivity, two approaches were investigated to determine the approximate derivatives in Eq. (5). The first approach was to apply finite differences with high order of precision. In the second approach, differentiable regression curves for the spatial and temporal progression of temperature were used. The finite differences approach, being the most straight-forward, collapses for calculation of ${\partial }^2\theta /\partial x^2$. It is due to limited number of temperature points and high thermal gradient in the direction of heat flow. The experimental data show some noise, which was amplified when calculating the derivatives. This is avoided in the second approach. The use of regression curves makes the temperature function and its derivatives continuous and smooth. Hence, the second approach is used and illustrated hereafter.

At each time step, at selected points in the specimen, the temperature is measured using thermocouples. Seven thermocouples labelled T1-T7 are used in the following tests. To calculate, four types of regression curves are initially selected: 2^nd order polynomial, 3^rd order polynomial, 4^th order polynomial, and exponential function. Based on the equation of each regression curve, the second derivative is determined. Exemplary results for the time t = 4000 s are shown in Figure 4 and Table 2. Although the results presented in Figure 4 and Table 2 are derived from a single time step of a specific experiment (Test 2) and at specific node (T2), they have been intentionally chosen as representative examples. The issues of instability or temperature fluctuations observed in these data are common but should be considered in a broader context.

Figure 4. Regression curves for time t = 4000 s.

Figure 4. Regression curves for time t = 4000 s.

The equations of the regression curves presented in Figure 4 are given in Table 2. The values of the second derivative vary significantly depending on the type of regression curve used. The use of polynomials for regression, as shown in Figure 4, can result in an inflection point in the analyzed range. As long as the specimen heats up on one side, the existence of an inflection point is physically incorrect. Thus, the inflection points seen in Figure 4 are related to experimental measurement noise. With this in mind, the exponential function was chosen as the one that gives the most reliable temperature profile inside the specimen. In exponential regression, the value of the second derivative is always of the same sign (no inflection points of the regression curve). In addition, Eq. (3) is a hyperbolic partial differential equation that can be solved analytically in some simplified cases, and the analytical solution is often in the form of an exponential function [19,20].

Table 2. Exemplary calculation of derivatives at selected time step and selected point.

Table 2. Exemplary calculation of derivatives at selected time step and selected point.

2nd order polynomial regression	$\theta \left(x,t=4000\mathrm{s}\right)=-14924x^2-\mathrm{7982,7}x+\mathrm{481,37}$ ${\theta }^{\text{'}\text{'}}(x,t=4000\mathrm{s})=-14924$
	Node T2	${\theta }^{\text{'}\text{'}}(x=0.001\mathrm{m},t=4000\mathrm{s})=-14924$
	Node T3	${\theta }^{\text{'}\text{'}}(x=0.011\mathrm{m},t=4000\mathrm{s})=-14924$
	Node T4	${\theta }^{\text{'}\text{'}}(x=0.013\mathrm{m},t=4000\mathrm{s})=-14924$
	Node T5	${\theta }^{\text{'}\text{'}}(x=0.018\mathrm{m},t=4000\mathrm{s})=-14924$
	Node T6	${\theta }^{\text{'}\text{'}}(x=0.023\mathrm{m},t=4000\mathrm{s})=-14924$
3rd order polynomial regression	$\theta \left(x,t=4000\mathrm{s}\right)=\left(1\times {10}^7\right)x^3-485786x^2-3263.7x+478.1$ ${\theta }^{\text{'}\text{'}}(x,t=4000\mathrm{s})=60000000x-971572$
	Node T2	${\theta }^{\text{'}\text{'}}(x=0.001\mathrm{m},t=4000\mathrm{s})=-911572$
	Node T3	${\theta }^{\text{'}\text{'}}(x=0.011\mathrm{m},t=4000\mathrm{s})=-311572$
	Node T4	${\theta }^{\text{'}\text{'}}(x=0.013\mathrm{m},t=4000\mathrm{s})=-191572$
	Node T5	${\theta }^{\text{'}\text{'}}(x=0.018\mathrm{m},t=4000\mathrm{s})=108428$
	Node T6	${\theta }^{\text{'}\text{'}}(x=0.023\mathrm{m},t=4000\mathrm{s})=408428$
4th order polynomial regression	$\theta \left(x,t=4000\mathrm{s}\right)=\left(6\times {10}^9\right)x^4-\left(3\times {10}^8\right)x^3+\left(6\times {10}^6\right)x^2-43843x+494.32$ ${\theta }^{\text{'}\text{'}}\left(x,t=4000\mathrm{s}\right)=72000000000x^2-1800000000x+12000000$
	Node T2	${\theta }^{\text{'}\text{'}}(x=0.001\mathrm{m},t=4000\mathrm{s})=-911572$
	Node T3	${\theta }^{\text{'}\text{'}}(x=0.011\mathrm{m},t=4000\mathrm{s})=-311572$
	Node T4	${\theta }^{\text{'}\text{'}}(x=0.013\mathrm{m},t=4000\mathrm{s})=-191572$
	Node T5	${\theta }^{\text{'}\text{'}}(x=0.018\mathrm{m},t=4000\mathrm{s})=108428$
	Node T6	${\theta }^{\text{'}\text{'}}(x=0.023\mathrm{m},t=4000\mathrm{s})=408428$
Exponential regression	$\theta (x,t=4000\mathrm{s})={492.25e}^{-23.2x}$ ${\theta }^{\text{'}\text{'}}(x,t=4000\mathrm{s})=264948.64e^{-23.2x}$
	Node T2	${\theta }^{\text{'}\text{'}}(x=0.001\mathrm{m},t=4000\mathrm{s})=258872.58$
	Node T3	${\theta }^{\text{'}\text{'}}(x=0.011\mathrm{m},t=4000\mathrm{s})=205272.01$
	Node T4	${\theta }^{\text{'}\text{'}}(x=0.013\mathrm{m},t=4000\mathrm{s})=195964.98$
	Node T5	${\theta }^{\text{'}\text{'}}(x=0.018\mathrm{m},t=4000\mathrm{s})=174501.96$
	Node T6	${\theta }^{\text{'}\text{'}}(x=0.023\mathrm{m},t=4000\mathrm{s})=155389.67$

Regression is also used to provide the derivative of temperature over time. Figure 5 shows the graph of the time-temperature relationship for an example measurement point and the linear regression for this data. The regression corresponding to the time t = 4000 s is built for a 20 second time interval from t = 3990 s to t = 4010 s. In a similar way, for the time t = 4001 s it will be the interval from t = 3991 s to t = 4011 s. Using linear regression, for time t = 4000 s one obtains:

$$\theta \left(x=0.001\mathrm{m},t\right)=0.1786·t-210.53,$$

(6)

$$\partial \theta \left(x=0.001\mathrm{m},t\right)/\partial t=0.1786.$$

(7)

From the determined value of the derivative of temperature over time and the exponential regression results from Table 4, the thermal diffusivity is determined for the measurement points (e.g., T2) and for the specific time t = 4000 s, which corresponds to a specific temperature $\theta$. The results are presented in Table 3.

Figure 5. An example of the time-temperature relationship and the corresponding linear regression for a given measurement point T2 (x = 0.001 m).

Figure 5. An example of the time-temperature relationship and the corresponding linear regression for a given measurement point T2 (x = 0.001 m).

The same methodology used to calculate the thermal diffusivity at node T2 is applied to the other nodes T3, T4, T5, and T6. The results are presented in Table 3. for thermal diffusivity for all node T2-T6, and for the specific time t = 4000 s. The determined values of the thermal diffusivity correspond to the specified temperatures $\theta$.

Table 3. Thermal diffusivity calculation for t = 4000 s and given measurement points (T2-T6).

Table 3. Thermal diffusivity calculation for t = 4000 s and given measurement points (T2-T6).

Thermal diffusivity for t = 4000 s	$\alpha \left(\theta \right)=\frac{\partial \theta }{\partial t}/\frac{{\partial }^2\theta }{\partial x^2}$
	Node T2	$\alpha \left(x=0.001\mathrm{m},t=4000\mathrm{s}\right)=0.1786:258872.58=6.90\times {10}^{-7}$(m2/s)
	Node T3	$\alpha \left(x=0.011\mathrm{m},t=4000\mathrm{s}\right)=0.1765:205272.01=8.60\times {10}^{-7}$(m2/s)
	Node T4	$\alpha \left(x=0.013\mathrm{m},t=4000\mathrm{s}\right)=0.1750:195964.98=8.93\times {10}^{-7}$(m2/s)
	Node T5	$\alpha \left(x=0.018\mathrm{m},t=4000\mathrm{s}\right)=0.1644:174501.96=9.42\times {10}^{-7}$(m2/s)
	Node T6	$\alpha \left(x=0.023\mathrm{m},t=4000\mathrm{s}\right)=0.1521:155389.67=9.79\times {10}^{-7}$(m2/s)

Table 4. Description of performed tests.

Table 4. Description of performed tests.

Experiments	Heat rate	DURATIONS
Test 1, Sample A (density of 114 kg/m3)	10°C per 60s	5 h	Test 1: fresh sample Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.
Colling Sample A	Room Temperature	12h	Cooling to room temperature.
Test 2, Sample A (density of 114 kg/m3)	10°C per 60s	5 h	Test 2: sample after Test 1 Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.
Test 3. Sample B (density of 114 kg/m3)	10°C per 60s	5 h	Test 3: fresh sample Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.
Colling Sample B	Room Temperature	12h	Cooling to room temperature.
Test 4, Sample B (density of 114 kg/m3)	10°C per 60s	5 h	Test 4: sample after Test 1 Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.
Test 5, Sample C (density of 114 kg/m3)	10°C per 60s	5 h	Test 5: fresh sample Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.
Colling Sample C	Room Temperature	12h	Cooling to room temperature.
Test 6, Sample C (density of 114 kg/m3)	10°C per 60s	5 h	Test 6: sample after Test 1 Heating from 21°C to 750°C during a period of 1h 20min = 4400 s, then a constant temperature of 750°C for the rest of duration.

3.4. Reduction of 3-D problem to 1-D problem

The assumption of 1-D heat transfer in the analysed region is verified numerically. To ensure that the diffusion of the heat is predominantly within the x-direction, it is necessary to analyse the derivatives of the heat flux in all directions in coordinates related to real positions of thermocouples and confirm that the derivative of the heat flux in the x-direction is significantly greater than the derivatives of the heat fluxes in the y- and z-directions.

The 3-D model of the heat transfer according to the test method described above is developed in ABAQUS software. The cross-sectional view of the model is given in Figure 6. To analyse the heat transfer problem, DC3D8 (8-node linear brick) finite elements were applied. The material property, i.e., the thermal diffusivity, which depends on temperature, is based exactly on the results of this paper. Chronologically, this analysis was done at the end, however, for the clearness of this paper, it is presented here. In the numerical model, the tie constraint is used between the specimen surfaces and the surfaces of the insulation and brick. The thermal boundary conditions are defined as heat fluxes due to convection and radiation on all surfaces. The thermal boundary condition is applied at the surfaces that don’t have the contact with the specimen, such as the external surfaces of brick element, the external surfaces of insulation, and also the top surface of the specimen. The temperature defined on the heated surfaces is equal to the one captured inside the furnace during the test. The initial temperature is predefined as 21°C.

Figure 6. Cross sectional view – nodal temperatures for t = 53 min, at the temperature of heated surfaces 535 °C.

Figure 6. Cross sectional view – nodal temperatures for t = 53 min, at the temperature of heated surfaces 535 °C.

Figure 7 presents the magnitude of the heat flux components (x, y and z) for points T1-T7. The results were obtained from numerical analysis. It is clear from the illustration below that the amount of the heat transfer in the x-direction is much greater than the heat flux in the y- and z-direction. However, the comparison of the expressions $\frac{\partial }{\partial x}\left({\lambda }_x\frac{\partial \theta }{\partial x}\right)$, $\frac{\partial }{\partial y}\left({\lambda }_y\frac{\partial \theta }{\partial y}\right)$ and $\frac{\partial }{\partial z}\left({\lambda }_z\frac{\partial \theta }{\partial z}\right)$ is needed to show that the 1-D heat flow assumption is close to reality.

Figure 7. Numerical result of the heat flux on x-, y- and z-directions of the specimen tested.

Figure 7. Numerical result of the heat flux on x-, y- and z-directions of the specimen tested.

To compare the partial derivatives $\frac{\partial }{\partial x}\left({\lambda }_x\frac{\partial \theta }{\partial x}\right)$, $\frac{\partial }{\partial y}\left({\lambda }_y\frac{\partial \theta }{\partial y}\right)$ and $\frac{\partial }{\partial z}\left({\lambda }_z\frac{\partial \theta }{\partial z}\right)$, the numerical model was applied in which nodes are evenly spaced in each spatial direction (distances between nodes are: $∆x=0.02$ m, $∆y=0.02$ m and $∆z=0.02$ m). The terms in the brackets represent the components of heat flux (e.g., ${\lambda }_x\frac{\partial {\theta }_i}{\partial x}=-q_{x_i}$). To determine the increments of these components in node i, the heat flux values $q$ in the neighboring nodes and the finite difference method were used. For the x-direction it is

$$\frac{\partial }{\partial x}\left({\lambda }_x\frac{\partial {\theta }_i}{\partial x}\right)=\frac{-q_{x_{i+1}}+q_{x_{i-1}}}{2∆x},$$

(8)

where $q_{x_{i+1}}$ and $q_{x_{i-1}}$ are the heat fluxes in the x-direction for the nodes i + 1 and i - 1, respectively.

Figure 8 presents the first derivatives of the heat flux according to the x-, y-, and z-directions for positions T1 and T2. It is clear that the derivative of the heat flux in the x-direction is one order of magnitude greater than in the other directions. The slight deviation observed in the z-direction results from the fact that a brick made of aerated concrete adheres to the tested specimen, which has a higher thermal conductivity than mineral wool. In future research, this shortcoming can be easily removed by inserting a layer of mineral wool between the brick and the specimen. Based on the presented analysis, it can be concluded that the assumption of a 1-D heat flow is close to reality.

Figure 8. The comparison of the 1^st derivatives of the heat flux at nodes T1 and T2 obtained from numerical analysis.

Figure 8. The comparison of the 1^st derivatives of the heat flux at nodes T1 and T2 obtained from numerical analysis.

4. Experiments

5. Results

5.1. Experimental results

The experiments carried out consisted in recording the temperature at the measurement points (T1-T7), where the temperature was a function of time. Figure 9 shows the recorded temperature graphs for samples A, B and C. Figure 9a,c,e correspond to Test 1, Test 3 and Test 5 in which fresh samples were tested. In these graphs, the disturbance appears between t = 3500 s and t = 5500 s. The disturbances are the result of the combustion reaction of the mineral wool binder, which causes an increase in the temperature inside the samples. In the second heating cycle of the samples (Test 2, 4 and 6) such a phenomenon was not noticed. It can therefore be concluded that the process of the sample preheating largely eliminates the heat generation inside the sample during Tests 2, 4 and 6.

Figure 9. Time-temperature dependence for the tested samples: (a) Sample A – first heating cycle; (b) Sample A – second heating cycle; (c) Sample B – first heating cycle; (d) Sample B – second heating cycle; (e) Sample C – first heating cycle; and (f) Sample C – second heating cycle.

Figure 9. Time-temperature dependence for the tested samples: (a) Sample A – first heating cycle; (b) Sample A – second heating cycle; (c) Sample B – first heating cycle; (d) Sample B – second heating cycle; (e) Sample C – first heating cycle; and (f) Sample C – second heating cycle.

All tests were performed for the same rate of temperature increase in the furnace. Nevertheless, it turned out that the phenomenon of binder burning occurred definitely the latest (after about 1 hour and 30 minutes) in sample C (Test 5), i.e. a sample in which the fibers are parallel to the plane affected by the temperature. In samples A and B, the fibers are perpendicular to the heated plane, which results in faster heat flow and earlier combustion of the mineral wool binder (after about 1 hour from the start of the test). Figure 9 conclusively demonstrate that the orientation of fibers significantly impacts the heat transfer characteristics within mineral wool. Hence, also the arrangement of fibers affects the pathways for heat conduction, convection, and radiation, influencing how heat is distributed and dissipated through within the material.

5.2. Thermal diffusivity

As it was mentioned in section 3.3, to determine the thermal diffusivity, it is necessary to calculate the second derivative of the temperature with respect to the spatial coordinate. The possibility of using a second-order polynomial, a third-order polynomial, a fourth-order polynomial and an exponential function was explored. The respective results for Sample A, Test 2, for measurement points T2-T6 are presented in Figure 10. The curves clearly illustrate significant changes in the results of the regression curve. In the case of polynomials of the second and third degree, a large variability of the second derivative is obtained. To obtain a good prediction of thermal diffusivity, the exponential regression was chosen for further investigation. Calculations of the derivative for each recorded time t, for points T2-T6, were performed using the MATLAB software.

Figure 10. The second derivative of temperature with respect to the spatial coordinate, obtained for Test 2, sample A using the regression curve function: (a) 2^nd order polynomial; (b) 3^rd order polynomial; (c) 4^th order polynomial; and (d) exponential function.

Figure 10. The second derivative of temperature with respect to the spatial coordinate, obtained for Test 2, sample A using the regression curve function: (a) 2^nd order polynomial; (b) 3^rd order polynomial; (c) 4^th order polynomial; and (d) exponential function.

A linear regression function determined for a time interval of 20 seconds was used to determine the temperature derivative with respect to the time variable (see Section 3.3). MATLAB software was used to calculate these derivatives for each recorded time t_i, for points T2-T6.

Following the completion of the derivative calculations, the thermal diffusivity values for points T2-T6 were determined using Eq. (5). The thermal diffusivity results for the second heat cycle of samples (A, B, and C) are presented in Figures 11–13. These figures illustrate the variation of thermal diffusivity as a function of temperature, providing valuable insights into the heat transfer properties of the materials at different temperature levels. Some differences can be noticed in the results obtained for the same temperature but for different measurement points (T2 to T6).

Figure 11. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5 and T6 – Sample A, Test 2.

Figure 11. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5 and T6 – Sample A, Test 2.

Figure 12. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5, and T6 - Sample B, Test 4.

Figure 12. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5, and T6 - Sample B, Test 4.

Figure 13. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5, and T6 - Sample C, Test 6.

Figure 13. Thermal diffusivity as a function of temperature for nodes T2, T3, T4, T5, and T6 - Sample C, Test 6.

Figure 14 shows the average value of thermal diffusivity from the curves obtained for measurement points T2-T6. One curve corresponds to one test (Test 2 - data from Figure 11, Test 4 - data from Figure 12 and Test 6 - data from Figure 13), each of these tests was performed on pre-heated samples. This is very important to eliminate internal heat sources before performing the test; calculations performed on data for samples without pre-heating lead to different results. The averaged functions shown in Figure 14 represent the general characteristics of the behavior of mineral wool subjected to temperature treatment. The results obtained for samples A and B are very similar (especially in the range up to 450°C), which means that the results are characterized by a certain reproducibility. The thermal diffusivity obtained for sample C is higher in the entire temperature range than for the other samples, which once again confirms the influence of fiber orientation on the thermal properties of mineral wool.

Figure 14. Thermal diffusivity (average value) as a function of temperature.

Figure 14. Thermal diffusivity (average value) as a function of temperature.

6. Validation procedure

Figure 15. 1-D model of heat transfer.

Figure 15. 1-D model of heat transfer.

Figure 16. Comparison of the experimental and numerical results: (a) Boundary conditions for the numerical problem - temperatures at nodes T1 and T7; (b) Time-temperature relationship at node T2; (c) Time-temperature relationship at node T3; (d) Time-temperature relationship at node T4; (e) Time-temperature relationship at node T5; and (f) Time-temperature relationship at node T6.

Figure 16. Comparison of the experimental and numerical results: (a) Boundary conditions for the numerical problem - temperatures at nodes T1 and T7; (b) Time-temperature relationship at node T2; (c) Time-temperature relationship at node T3; (d) Time-temperature relationship at node T4; (e) Time-temperature relationship at node T5; and (f) Time-temperature relationship at node T6.

7. Conclusions

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