**1. Introduction**

Thermal conductivity is a physical property of materials that measures heat conduction capacity. In other words, thermal conductivity is also the ability of a substance to transfer the kinetic energy of its molecules to adjacent ones or to substances with which it is in contact. In the International System of Units, the thermal conductivity is measured in W/(m K) equivalent to J/(m s K). There are several methods to measure the thermal conductivity of materials: the most conventional method for measuring thermal conductivity consists of two concentric metal spheres, of very small thickness to minimise the heat capacity of the system. It has not been used with measurements greater than

300 ◦C [1]. The parametric study consists in obtaining the temperature distribution in the most insulating composite bar system for different operating conditions. As operating conditions, it refers to the temperature difference at the ends of the system, characteristics of the reference material, aspects of the insulating material and features of the sample materials [1,2]. The determination of the thermophysical properties of materials is essential in all processes where energy exchanges occur, in particular, heat. For the design, operation and maintenance of systems and equipment where the temperature is present, it is essential to know the value of these properties in particular of thermal conductivity. This property has an important effect on solid thermal conductive materials such as aluminium, iron, copper, its alloys, and new materials that are used to build equipment and machinery parts, such as automotive vehicle engines [1]. Thermal conductivity is also an issue related to the second law of thermodynamics or the law of entropy, which governs most of the phenomena that occur in the universe, by which it is estimated that any process that involves work increases the entropy of the universe (increases the disorder and chaotic movement of atoms and the temperature of existing molecules and grains). Thermal energy always flows spontaneously from highest to lowest concentration, or from hot to cold. This implies that heat transfer by conduction occurs from one body to another at a lower temperature or between areas of the same material but with a different temperature. Heat transmission involves an internal energy exchange, which combines potential energy and kinetic energy of electrons, atoms, and molecules: the higher thermal conductivity, the better the heat conduction. The inverse property is the thermal resistivity, which indicates that, at lower thermal conductivity, more heat insulation (more resistivity). Concerning potential energy, we can say that it is the mechanical energy associated with the location of a body in a field of forces or the presence of an area of effects within the body itself. The potential energy is the result that the system of forces that affects a given body is conservative, then, the total work on a particle is zero. The kinetic energy of a body, meanwhile, is what it has thanks to its movement. It is the work needed to achieve its acceleration from rest to a given speed. When the body reaches this energy throughout the acceleration, it maintains it unless it alters its speed. To return to the resting state, it is necessary to perform a dangerous job with the same magnitude. By heating matter, the average kinetic energy of its molecules increases, and this increases its level of agitation. At the molecular level, heat conduction occurs because the molecules interact with each other by exchanging kinetic energy without making global movements of matter. It should be mentioned that at the macroscopic level, it is possible to model this phenomenon by means of Fourier's law.

The CENAM implemented a system to measure thermal conductivity of solid conductive materials, and the design criteria were developed for the construction of the measurement system, which operates under the condition of heat flow in a permanent state. The system uses a reference material, which limits the accuracy of the method. An analysis of the system is carried out considering that there is axial and radial heat flow. In addition, the solid bar of material that can be evaluated, it has a hollow bar of insulating material. The problem to be solved is a bar composed of a reference conductor material at the longitudinal ends, and a test material depicted in the centre, the entire bar an element by an insulating material, it is considered that it is axial and radial flow and the physical dimensions of the problem are shown in Figures 1a, 2 and 3. Using the apparatus developed in the CENAM, two concentric cylinders are used, housing the material to be tested between them. Inside the smaller diameter cylinder is placed the heating resistance, which is covered with another cylinder to standardise the surface temperature. The temperature measurement is carried out on the outer and inner cylinders, using thermocouples for this. The method is used to measure thermal conductivity in materials solid conductors. To meet this need for measurement, the CENAM developed a system for measuring thermal conductivity in thermally conductive solid materials employing a secondary method. This work presents a comparison of certain experimental results using the cut bar and the finite element method (FEM), to obtain information that serves in the development of the new cut bar system, to extend its operating range up to 600 ◦C, under optimal operating conditions [2].

**Figure 1.** (**a**) Schematic of a Comparative-Guarded-Longitudinal Heat Flow System, indicating possible locations of temperature sensors (**b**) methodology for the experiment used in this work [1].

**Figure 2.** Cut bar method system [2].

**Figure 3.** Diagram of the experimental model used.

## **2. Technique Background**

The method determines the thermal conductivity of a sample using a reference material by a permanent state technique known as the concentric bar cut method [3]. The system consists of a bar with well-known properties, called reference bar, another bar with conductivity to be determined, called sample o test bar, and another reference bar. The composite metal bar is covered with an insulating material to prevent heat flow in the radial direction. At one end of the composite bar, a heat source is placed, and at the opposite end, there is a heat sink or cold source [4,5]. Then, employing temperature and length measurements, the conductivity of the sample material can be determined. Figure 1 shows a diagram of the composite bar system.

The arrangemen<sup>t</sup> diagram of the bars for the method used in this work is shown in Figure 1a. The reference bars are located at the ends and the test bar in the center of both. At one end of the bar array, a heat sink or cold source is located; at the other end, a heater that allows generating a temperature gradient, necessary for the determination of the thermal conductivity value. Marked with *x* in Figure 1a is where the thermocouples indicated at a certain height in mm and designated according to the letter *z* are located; the thermocouples type T were a fine wire of 0.6 mm diameter from OMEGA brand, calibrated by CENAM. *rA* indicates the radius of the bars used for the test and *rB* the radius of the insulation used in the test. A force is applied axially to improve the contact between the bars axially.

A brief description of the process performed to carry out the test is named in Figure 1b. Before starting the test the bars to be used need to meet the necessary diameter and height, as well as some flatness on the flat faces of the bars, after these perforations are made on the cylindrical face a few millimeters deep to house the thermocouples later. After finishing the bars, they are placed one above the other in the order, as appear in Figure 1a. Then the thermocouples are placed in each one of the sweepers made in the bars, then it is surrounded with the insulating material, and the guard is added; the axial force is applied to improve the contact. The next step is to adjust the operating temperature of the hot and cold source according to the measurement temperature at which the test is required to reach the operating temperature was used as a power supply, which supplies the necessary voltage to an electrical heater until it reaches temperature operation, which is registered by the thermocouple located in the hot source. For example, if a test temperature of 100 ◦C is required, the average temperature of the hot and cold source must be sought to be 100 ◦C. For example, the temperature of the hot source at 150 ◦C and the cold source at 50 ◦C, so the average temperature is 100 ◦C. In this way, several combinations can be generated. Once the operating temperatures have been adjusted, data acquisition begins, by a program in LabView developed by CENAM, where the signal of thermocouples are read by a multimeter and send to a PC to register its values. The values will be adequate when a the steady-state regime has been reached, it is known, because charts of temperature from thermocouples do not change, reach a constant temperature throw experiment in time. With the acquired data, Equation (1) is used to calculate the thermal conductivity value.

From the work in [2] it was found that the thermal conductivity of the sample is given by

$$
\lambda\_M = \frac{Z\_4 - Z\_3}{T\_4 - T\_3} \left[ \frac{\lambda\_{R\_1}}{2} \left( \frac{T\_2 - T\_1}{Z\_2 - Z\_1} \right) + \frac{\lambda\_{R\_2}}{2} \left( \frac{T\_6 - T\_5}{Z\_6 - Z\_5} \right) \right] \tag{1}
$$

where *λ M* is the thermal conductivity of the sample. *λR*1, and *λR*2 are the thermal conductivity of reference materials 1 and 2. *Ti* is the temperature in each of the *Zi* positions where the thermocouples are placed. Subscripts 1 and 2 refer to the first reference bar, 3 and 4 to the sample under measurement and 5 and 6 to the second reference bar.

If the distances between the thermocouples of each bar are equal and the reference material is the same for the two bars, so from Equation (1) which the thermal conductivity of reference materials leave the equation as a common term. Taking into account that the distances are also equal, (*<sup>Z</sup>*2 − *<sup>Z</sup>*1) is the same that (*<sup>Z</sup>*6 − *<sup>Z</sup>*5) then leave the parenthesis so with (*<sup>Z</sup>*4 − *<sup>Z</sup>*3) obtain unity. Then, to simplify is rewritten (*<sup>T</sup>*4 − *<sup>T</sup>*3) such as Δ *T*2, rewrite (*<sup>T</sup>*6 − *<sup>T</sup>*5) such as Δ *T*3 and (*<sup>T</sup>*2 − *<sup>T</sup>*1) such as Δ *T*1. Then, Equation (1) is reduced to

$$
\lambda\_M = \frac{\lambda\_{R\_2}}{2} \left( \frac{\Delta T\_1 + \Delta T\_3}{\Delta T\_2} \right) \tag{2}
$$

where *<sup>λ</sup>R*2 is the same the *<sup>λ</sup>R*1 because the reference material are equal. Δ *T*1 and Δ *T*3 are the difference among each reference bar and Δ *T*2 is the difference of temperature of the test bar. The cold source or heat sink is constituted by a 10 cm diameter copper plate that has a 10 mm diameter copper tube coil welded through which a fluid such as ethylene glycol flows from a bath of controlled temperature. One of the surfaces is in contact with one end of a reference bar and the other part in an insulated container [6]. The recirculation bath can maintain the temperature of the cold source between −30 ◦C and 60 ◦C.

The reference material bars are 99.999% high purity copper with a diameter of 19.1 mm and a length of 60 mm. The composite bar is surrounded by a 100 mm diameter polyvinyl chloride (PVC) pipe, the inside of which contains 50.8 mm thick fiberglass.

The measurement system has seven calibrated type T thermocouples. The electromotive force (EFM) of each thermocouple is measured with a digital high-accuracy multimeter of 81 2 digits model 3458A from Agilent Technologies and aided by an 8-channel scanner keithley 7001 both manufactured in USA. CENAM developed a computer program for the control, reading, and recording of data. The program to acquire data works with a graphic interface developed in LabView, where is registered tension measurement of each thermocouple and through of coefficient obtained from calibration and with the Newton–Raphson method is converted tension measurement to a temperature value. The EFM was measurement by scanner and multimeter connected to a PC. That value is introduced in a subprogram, where is converted to each temperature value from each thermocouple used in the experiment. With the distance between thermocouples of each bar and the thermal conductivity from reference bar is calculated the thermal conductivity of the bar under test. In the front panel of Labview developed by CENAM, are showed temperature of each thermocouple used, temperature of hot and cold source, constants from calibration of thermocouples used, graphs of Δ *T*1, Δ *T*3 and Δ *T*2 as well as the thermal conductivity value. Figure 2 shows a schematic of the measurement system [7].
