# Thermophysical Property Measurements with the Finite Bar
## 3 Experimental Setup
### Correction of Systematic Errors  
For the geometric difference, the bar is separated into the heating cartridge (H), the hollow cylinder (C) and the bar (B). In the small Fourier-Number limit, it must be ensured that the characteristic time scales \(\tau\) of the heating cartridge with diameter \(d_{H}\) and the hollow cylinder are small compared to the characteristic timescale of the bar.  
\[\begin{array}{rcl}\tau_{H}=\frac{d^{2}}{16a_{H}}\ll&\tau_{B}=\frac{L^{2}}{a _{S}}\\ \tau_{C}=\frac{(d_{x}-d_{i})^{2}}{4a_{S}}\ll&\tau_{B}=\frac{L^{2}}{a_{S}}\\ \end{array} \tag{17}\]  
This requires the knowledge of the thermal diffusivity (\(a\)) of the sample (\(S\)) material and the cartridge for the first condition and can only be established a posteriori. The second condition is easily verified when the wall thickness of the hollow cylinder is significantly smaller than the length of the bar. The heating of the cartridge \(\dot{H}_{H}\) and the heating of the  
Figure 3: Temperatures at the different locations of the bar, \(T_{1}\) is next to the heating cartridge, \(T_{7}\) at the adiabatic end of the bar.  
hollow cylinder \(\dot{H_{C}}\) can then be subtracted from the electric heating power \(P_{el}\). The diabatic contribution should remain small as the temperature difference to the ambient is small during this phase.  
The Enthalpy change of the heating cartridge and the hollow sample cylinder at the end can be simply estimated by the temperature change.  
\[\dot{H_{H}}=m_{H}c_{p,H}\frac{\partial T_{H}}{\partial t}\qquad\dot{H_{C}}=m_{C} c_{p,C}\frac{\partial T_{C}}{\partial t} \tag{18}\]  
In the large Fourier-Number limit, when the asymptotic temperature profiles are established, the enthalpy balance of the experimental setup can be established.  
\[\dot{H_{H}}+\dot{H_{C}}+\dot{H_{B}}+\dot{Q_{L}}+P_{el}=0 \tag{19}\]  
The heat loss due to the diabatic condition \(\dot{Q_{L}}\) can be modeled by assuming a convective heat transfer to the ambient temperature \(T_{A}\).  
\[\dot{Q_{L}}=\alpha A(T_{A}-T_{B}) \tag{20}\]  
The necessary coefficient for the convective heat transfer \(\alpha A\) can be extracted from the cooling phase with \(P_{el}=0\) W. When the energy is equally distributed within the bar, all temperature sensors show the same temperature. Then the bar can be considered as a thermal block capacity and the analytic expression for the temperature decrease \(T_{cool}(t)\) can be fitted to the data to estimate the time constant \(\tau\).  
\[T_{cool}(t)=T_{a}+(T_{0}-T_{a})e^{-(t-t_{0c})/\tau} \tag{21}\]  
Here \(T_{a}\) is the ambient temperature surrounding the block and \(T_{0}\) is the temperature at the starting time \(t_{0c}\) of the block capacity cooling process. Using the estimated heat capacity \(c_{p}\) and the relation for the characteristic time \(\tau=(mc_{p})/(\alpha A)\), the heat flux to the surrounding of the bar can be estimated by  
\[\dot{Q}=\frac{mc_{p}}{\tau}(T_{cool}(t)-T_{a}) \tag{22}\]  
Measurements with insulating materials such as ceramic fiber insulation (thermal conductivity \(\lambda\) of 0.07 W/mK) and different heating power revealed an unwanted effect of transient heating of the insulation material with an undefined heat loss compared to free convection. Therefore, it is easier to compensate for the free convection heat loss than compensating for the retained heat in the non-ideal insulation.