# Thermophysical Property Measurements with the Finite Bar
## 1 Introduction  
The efficient design of cooling structures with modern manufacturing technologies such as 3D metal printing requires the knowledge of the thermophysical properties of the printed material [1]. Different methods for the determination of thermophysical material properties of solid materials are available. Most of the methods are based on temperature measurement and comparison to analytical solutions [2] of the temperature. The methods differ in the sample geometry and the form of applying energy to the sample. The methods can be classified accordingly into an energy pulse as in the laser flash method, fixed boundary temperatures or fixed heat flux at boundaries. In the laser flash method [3], a sample of the material is exposed to a laser flash on one side and the temperature rise of the probe is used to determine the thermal diffusivity, see [PERSON] et al., 1970 [4] and the corresponding standards and references (ASTM E2585-09 [5], ASTM E1461-13 [6]). In the Sears bar method [7], a linear sample of the material is kept at fixed temperature at both ends. The temperature and the applied heat are measured. This allows to determine the thermal conductivity. In the transient line source method [8], a probe of small diameter is inserted into a cylindrical sample of much larger diameter. The applied heat and the temperature are monitored over time. The logarithmic temperature rise allows determining the thermal conductivity. In this paper, a rather simple method for the determination of thermophysical properties based on one dimensional conduction is presented. It differs from the previously mentioned methods by a cost-effective realization without any potentially dangerous equipment like a laser and no necessity of a regulated heat sink. The boundary conditions are defined by a constant heat flux at one side of the bar and a diabatic condition at the other end of the bar. Monitoring the temperature at different locations and comparison to the analytical solution for small and large Fourier numbers allows the determination of thethermophysical properties. In the literature, analytical solutions for the one dimensional bar are available for several boundary conditions, see [PERSON] [2; 9]. The time dependent analytical spacial temperature profile is then typically given by an infinite series. An alternative and mathematically simple way is to restrict oneself to the asymptotic solutions of very small and very large physical time scales. In the limit of small Fourier-Numbers at small physical time scales when the heat flux has not caused a temperature rise at the adiabatic end, the analytical solution of the semi-infinite body can be applied. The temperature evolution can then be used to determine the product of thermal conductivity, density and heat capacity. In the case of large Fourier-Numbers, a quasi-steady spacial temperature profile develops. It is only shifted by an offset temperature. The change of the offset temperature is then used to determine the heat capacity of the sample material and the spacial shape of the temperature curve allows determining the conductivity. The theoretical considerations are only directly applicable to the measurement when adiabatic boundary conditions are realized. As the reality is diabatic the heat flux to the surrounding can be partially compensated by measuring the temperature while the sample is cooling under the same boundary conditions as it was heated. Compared to other methods, this method has the advantage of allowing to determine several thermophysical properties, i.e., heat capacity \(c_{p}\) and conductivity \(\lambda\) in one measurement for a wide range of conductivities. In the case of low conductivities, the small Fourier-number limit, the thermal diffusivity \(a\) can be determined and in the cases of large conductivities, the large Fourier-number limit, heat capacity \(c_{p}\) and conductivity \(\lambda\) can be determined.