# Theoretical Description of Carbon Felt Electrical Properties Affected by Compression
## 3 Interpretation of Experimental Data
### Analytical Model of CF conductivity  
The electrical conductivity of CF is achieved with carbon (graphite) filaments. More detailed analysis of the CF electrical conductivity leads to a rigorous conclusion that only those carbon filaments which simultaneously contact both electrodes of the cell sustain electron flow. Therefore, CF conductivity may be assessed as a sum of individual filaments conductivity comprising the total current flow between electrodes. Any individual random disposition of filaments in a CF structure (Figure 7 could be represented by a diagonal of parallelepiped ABCDA'B'C'D' as in Figure 7. For simplicity, individual fibers are considered as thin rods having similar diameters, and therefore, equal cross-sections. This circumstance follows from the data in Table 1. The edge CC\({}^{*}\) in the parallelepiped is equal to CC\({}^{*}=L\cdot\sin\varphi\) and symbolizes a height of a filament having length L and polar disposition angle \(\varphi\). In accordance with the working hypothesis, only those filaments contacting both electrodes are participating in current flow; therefore, they must have heights greater than the width (denoted  
Figure 6: The relative increase of filaments interconnections versus relative volume decrease. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\varepsilon_{\mathbf{r}}\)} & \multirow{2}{*}{**Liquid**} & \multicolumn{2}{c}{**K\({}_{\text{incr}}\) = A(e\({}^{\mathbf{B}\cdot\mathbf{D}\cdot\mathbf{1}}\)) + 1**} & \multirow{2}{*}{**coeff. R\({}^{2}\)**} \\ \cline{3-3} \cline{5-5}  & & **A** & **B** & \\ \hline
1 & dry & 5.499 & 0.03042 & 0.991 \\
42.5 & glycerol & 2.832 & 0.0322 & 0.98 \\
21.6 & ethanol & 0.5523 & 0.05052 & 0.996 \\
2.02 & cyclohexane & 0.8402 & 0.03486 & 0.986 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Coefficients in approximating function (5) defining a relative increase of filaments interconnections during felt compression. as W) of the CF. Thus, the totality of fibers whose edge CC\({}^{*}>=\) W contribute to the total electrical conductivity of CF. Considering a stochastic length and disposition of filaments in the space of the CF frame, the summing of conductivities can be performed only by a probabilistic approach. Firstly, a stochastic distribution of their lengths (Figure 8a) was measured by a microscope and described by truncated Gauss function. The appropriate parameters were found to include the average length of L\({}_{\text{av}}\) = 25 mm, with a standard deviation (STD) of 11 mm. The measurement of the angle \(\varphi\) presents a significant technical problem. It was suggested to measure horizontal (\(\alpha\)) and vertical (\(\beta\)) angles separately by the same optical microscope approach. The histograms of their distributions are shown in Figure 8b,c. In addition, for the simplification of succeeding calculations, it was assumed that both these angles and their distributions are independent. Knowing the horizontal and vertical angles allows calculating the polar angle \(\varphi\) as:  
\[\varphi=\arccos\left(\frac{\cos\beta}{\sqrt{\cos^{2}\alpha+\sin^{2}\alpha\cdot \cos^{2}\beta}}\right) \tag{6}\]  
Considering the distribution of horizontal and vertical angles obtained in optical measurements, a Gamma function was selected to describe their probabilities. The expression of the Gamma function (7) includes two parameters (a and b) which determine an exact curve of a probability function p of a variable x, it's mean and standard deviation (STD) values. \[p(x|a,b)=\left\{ \begin{array}{ll}\frac{1}{\mu\Gamma(a)}x^{a-1}e^{-\frac{x}{b }},&x\geq 0\\ 0,&x<0\end{array} \right., \tag{7}\]  
Figure 8: (**a, b**) Illustration of the horizontal (\(\alpha\)) and vertical (\(\beta\)) angles disposition inside carbon frame. Figure 7: Disposition of individual fiber in the CF frame. The horizontal angle \(\alpha\) distribution is described by parameters a = 1 and b = 22. Whereas for angle \(\beta\): a = 1.4, b = 20. Stochastic distributions of polar angles are illustrated in Figure 8a,b.