# Non-Destructive Infrared Evaluation of Thermo-Physical Parameters in Bamboo Specimens
## 2 Theoretical Model  
For this research, we use the Cepa of the Guadua a.k, which corresponds to the lowest part of the trunk. The wall thickness and outside diameter of Cepa provide appropriate strength for its use in construction [22]. Cepa geometry can be properly described with cylindrical coordinates. Therefore, in this section, we present the theoretical model of the radial heat transfer in a cylinder heated from inside to a temperature \(T_{s}(t)\) and subjected to convective heat loss. This phenomenon is formulated in Equation (1). \[\frac{1}{r}\frac{\partial}{\partial r}\left(rk\frac{\partial T}{ \partial r}\right) =\rho c_{p}\frac{\partial T}{\partial t} \text{in }\ r_{0}<r<R, \text{for }t>0, \tag{1a}\] \[T =T_{s}(t) \text{at }\ r =r_{0}, \text{for }t>0,\] (1b) \[\frac{\partial T}{\partial r} =\frac{h}{k}(T-T_{\infty}) \text{at }\ r =R, \text{for }t>0,\] (1c) \[T =T(r) \text{for }t =0, \text{in }\ r_{0}<r<R, \tag{1d}\]  
where \(k\) is the thermal conductivity, \(\rho c_{p}\) is the volumetric heat capacity, \(T_{s}(t)\) is the temperature at \(r=r_{0}\), and \(T(r)\) is the cylinder initial temperature between the inner and outer walls. The Crank-Nicholson (CN) method solves numerically the heat transfer model presented in Equation (1). This method, besides providing unconditional stability, offers second order accuracy \(O(\Delta r^{2},\Delta t^{2})\) in both space and time [23; 24]. Before finding the CN representation of Equation (1a), let us rewrite it as follows:  
\[\frac{\partial^{2}T}{\partial r^{2}}+\beta\frac{\partial T}{\partial r}=\frac {1}{\alpha}\frac{\partial T}{\partial t}, \tag{2}\]  
where \(\alpha=\frac{k}{\rho c_{p}}\) is the thermal diffusivity and \(\beta=\left(\frac{1}{r}+\frac{1}{k}\frac{\partial k}{\partial r}\right)\) is an extra coefficient added to the equation when the thermal conductivity is assumed to vary with respect to \(r\). Even though here we express Equation (2) in its CN form to have a more general approximation, for the inversion procedure, we supposed \(k\), \(\rho c_{p}\), and \(h\) to be constant to avoid ill-conditioning in increasing the dimension of \(k\). Now, to find the finite difference equation as in the CN method, we substitute the following second order derivative approximations into Equation (2):  
\[\frac{\partial T}{\partial t}(r_{i},t_{l}) \approx\frac{T_{i}^{l+1}-T_{i}^{l}}{\Delta t}, \tag{3a}\] \[\frac{\partial T}{\partial r}(r_{i},t_{l}) \approx\frac{T_{i+1}^{l}-T_{i-1}^{l}}{2\Delta r},\] (3b) \[\frac{\partial^{2}T}{\partial r^{2}}(r_{i},t_{l}) \approx\frac{T_{i+1}^{l}-2T_{i}^{l}+T_{i-1}^{l}}{2\Delta r^{2}}+ \frac{T_{i+1}^{l+1}-2T_{i}^{l+1}+T_{i-1}^{l+1}}{2\Delta r^{2}}. \tag{3c}\]  
After plugging Equation (3) into Equation (2), we obtain Equation (4), where \(\lambda_{i}=\frac{\alpha_{i}\Delta t}{2\Delta r^{2}}\), \(\Gamma_{i}=\frac{\beta_{i}\Delta r}{2}\). \(\beta_{i}\) is expressed as \(\frac{1}{r_{0}+\Delta ri}+\frac{1}{k_{j}}\frac{\partial k}{\partial r}(r_{i})\). \(\frac{\partial k}{\partial r}(r_{i})\) is approximated by using second order forward and backward differences at the first and last points of the grid, respectively, while it is approximated by using center differences at the interior points:  
\[ \begin{array}{c}\lambda_{i}T_{i-1}^{l+1}-(2\lambda_{i}+1)\ T_{i}^{l+1}+ \lambda_{i}T_{i+1}^{l+1}=...\\ -\lambda_{i}\left(1-T_{i}\right)T_{i-1}^{l}-(2\lambda_{i}-1)\ T_{i}^{l}- \lambda_{i}\left(1+\Gamma_{i}\right)T_{i+1}^{l}.\end{array}  \tag{4}\]  
Using Equation (4), we can set up a system of \(n-2\) linear equations that allow us to solve for the temperatures \(T_{i},\ i=2,...,n-2\) for \(t_{l+1}\). The values \(T_{1}^{l}\) and \(T_{n}^{l}\) allow the incorporation of the boundary conditions (Equation (1b,c)) into the CN model by modifying Equation (4) for \(i=2\) and for \(i=n-1\)as indicated in Equations (5) and (6), respectively. In these two equations, l.b.c. and r.b.c. stand for a term added by the incorporation of the left and right boundary conditions, respectively:  
\[