# Thermophysical Property Measurements with the Finite Bar
## 2 Physical Foundations
### Solution in the Large Fourier-Number Limit  
At large physical time scales, when the Fourier-Number \(Fo=at/L^{2}\) is many times greater than 1, the temperature profile becomes quasi-steady with respect to space.  
The analytical solution can be derived, considering the heat source term \(\dot{Q}\) to be a Dirac function in space on the left side of the bar (\(\pi=0\)). This can also be viewed as a boundary condition on the left side, with \(\frac{\partial T}{\partial x}=-\frac{Q}{\partial\dot{X}}\) with \(\lambda\) being the unknown thermal conductivity and \(A\) the surface on the left side of the bar. The shape of the temperature profile from left to right remains similar, it is simply shifted by the increase in the average bar temperature. As the temperature difference \(\Delta T_{lf}=T(x=0,t)-T(x=L,t)\) from the left side of the bar \(T(x=0,t)\) to the right side of the bar \(T(x=L,T)\) remains constant, it decouples from the temporal change in temperature.  
\[T_{as}(x,t)=\frac{\dot{Q}\cdot t}{mc_{p}}+\Delta T_{lf}f(x) \tag{9}\]  
We refer to this temperature profile as the asymptotic temperature solution \(T_{as}(x,t)\) given by Equation (9). The time-dependent component is given by the constant power heat source \(\dot{Q}\). The spatial distribution of the temperature is time independent and represented by \(f(x)\). This leads to simple derivatives with respect to time and space. The temporal derivative is simply the heating power divided by mass and heat capacity. The second spatial derivative consists of the temperature difference \(\Delta T_{lf}\) times the second spatial derivative \(\frac{\partial^{2}f(x)}{\partial x^{2}}\).  
\[\frac{\partial T(x,t)}{\partial t}=\frac{\dot{Q}}{mc_{p}}\qquad\frac{\partial^ {2}T}{\partial x^{2}}=\Delta T_{lf}\frac{\partial^{2}f(x)}{\partial x^{2}} \tag{10}\]Placing the terms into the temperature diffusion Equation (2) yields the following relation.  
\[\frac{\dot{Q}}{mc_{p}}=a\Delta T_{IF}\frac{\partial^{2}f(x)}{\partial x^{2}} \tag{11}\]  
With a constant heat source \(\dot{Q}\), \(f(x)\) is a simple second order polynomial. For this reason, the asymptotic temperature distribution \(T_{as}\) must follow the following equation with unknown constant \(c_{1}\) and \(c_{2}\).  
\[T_{as}(x,t)=T(0,t)+c_{1}\cdot\left(\frac{x}{L}\right)+c_{2}\cdot\left(\frac{x }{L}\right)^{2} \tag{12}\]  
The unknown constants can be obtained with the boundary conditions on the left and the right end of the bar. Next to the temperatures at both ends of the bar, the lack of heat flux at the right end of the bar gives an additional condition for the spatial derivative of the temperature profile.  
\[\frac{\partial T(x,t)}{\partial x}\left|{}_{x=L}\right.=0 \tag{13}\]  
Applying the boundary condition leads to a simple second order polynomial for the temperature profile in the bar.  
\[T_{as}(x,t)=T(0,t)+\Delta T_{IF}\cdot\left(\frac{x}{L}\right)\left(\left( \frac{x}{L}\right)-2\right) \tag{14}\]  
Temperature measurements at both ends of the bar are sufficient to determine \(\Delta T_{IF}\) and to verify that the asymptotic state is reached. Then the temporal change in temperature can be used to determine the specific heat capacity \(c_{p}\).  
\[c_{p}=\frac{\dot{Q}}{m\frac{\dot{Q}}{\partial t}T(x=0,t)} \tag{15}\]  
The temperature difference \(\Delta T_{IF}=T(x=0,t)-T(x=L,t)\) at both ends of the bar allows determining the thermal diffusivity \(a\) in the large Fourier-Number limit and thereby the conductivity \(\lambda\).  
\[\begin{array}{ll}a&=\frac{\dot{Q}}{mc_{p}}\frac{\dot{L}^{2}}{2\Delta T_{IF}} \\ \lambda&=a\rho c_{p}\end{array} \tag{16}\]  
The obtained values can be compared to the values determined in small Fourier-Number limit.