## 2 Technique Background
\(T_{i}\) is the temperature in each of the \(Z_{i}\) 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, \((Z_{2}-Z_{1})\) is the same that \((Z_{6}-Z_{5})\) then leave the parenthesis so with \((Z_{4}-Z_{3})\) obtain unity. Then, to simplifyis rewritten \((T_{4}-T_{3})\) such as \(\Delta T_{2}\), rewrite \((T_{6}-T_{5})\) such as \(\Delta T_{3}\) and \((T_{2}-T_{1})\) such as \(\Delta T_{1}\). 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 \(\lambda_{R_{2}}\) is the same the \(\lambda_{R_{1}}\) because the reference material are equal. \(\Delta T_{1}\) and \(\Delta T_{3}\) are the difference among each reference bar and \(\Delta T_{2}\) 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 \(-30\)\({}^{\circ}\)C and \(60\)\({}^{\circ}\)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 \(8\frac{1}{2}\) 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 \(\Delta T_{1}\), \(\Delta T_{3}\) and \(\Delta T_{2}\) as well as the thermal conductivity value. Figure 2 shows a schematic of the measurement system [7].