# Theoretical Description of Carbon Felt Electrical Properties Affected by Compression
## 1 Introduction
The only difference could be observed in heterogeneous mathematical formulations for describing CF conductivity. ASR magnitude was characterized [26], for example, by an exponential function. Subsequent results of ASR versus CF compression coefficient were obtained [27]. Since CF could be bound to the surface of the electrode either by simple mechanical contact or by adhesives, the influences on ASR of these materials were investigated. Assemblies of different electrode plates were examined. The same sharp diminishing of resistance versus compression pressure was detected, which may be described by an exponential formula, although with different exponent coefficients. Predictably, these coefficients depend on the type of adhesive used; however, the significant achievement of [PERSON] et al. 's work [26] is that for the first time their values were represented as a function of the kind of adhesive and the type of electrodes design. More recent reports have focused on experimental measurements of CF conductivity [28; 29; 30] and have shown influences of felt porosity and specific resistivity of the carbon material. For example, the analytical expression of ASR (named as a specific electrical resistivity and denoted as \(\rho_{felt}\)[28]) was defined by the following expression:  
\[\rho_{felt}=\frac{4}{(1-\lambda)}\rho_{carbon}, \tag{1}\]  
where \(\lambda\) signifies relative felt porosity and is equal to the volume ratio of its empty space to the total felt. Expression (1) was found to be in good agreement with the experimental results. Experimental measurements of ASR for CF electrodes used in vanadium redox batteries were carried out in [32]. ASR, in this work, is denoted as R\({}_{\text{ASR}}\). R\({}_{\text{ASR}}\) with units of m\(\Omega\cdot\)cm\({}^{-2}\) was estimated by Equation (2) below, where V\({}_{\text{f}}\) denotes the percentage of fiber volume. Notably, V\({}_{\text{f}}\) is expressed by the porosity (\(\lambda\)), i.e., V\({}_{\text{f}}\)/100 = 1 - \(\lambda\). Then  
\[R_{ASR}=144.456\cdot\left(3-\frac{10}{V_{f}}\right)^{-1.117}, \tag{2}\]  
The exponential type of expression (2) is valid for various electrochemical applications. However, all the specific constants in Equation (2) are only applicable to the description of resistance of electrodes used in special vanadium redox batteries. The use of this expression in other applications will lead to considerable loss of accuracy. This is because an actual type of carbon material and the manufacturing process of CF were neglected in this work. Therefore, this circumstance significantly diminishes the value of the presented results. Furthermore, resistance measurements were carried out in dry conditions, i.e., in the air only. However, supplementary investigations performed by the authors [21] showed the effects of liquid dielectric constant (relative permittivity) on CF conductivity. The electrical conductivity of CF seems to be influenced by the properties of these liquids conducive to the penetration of fluid between fibers and their separation or conversely, to their convergence and additional contacts. The resistance of CF is known to increase through a diminishing interconnection of density of fibers. On the other hand, the resistance is diminished by additional interconnections [21]. It is possible to summarize that currently, many results regarding CF electrical conductivity have been obtained, processed, and described. However, a strict analytical explanation of the CF electrical resistance phenomena has not been propounded. This could be explained by the significantly stochastic structure of felt that was not considered in previous works. Known theoretical approaches used relatively simplified assumptions, reducing the applicability of their results. The requirement to find a more comprehensive analytical explanation of CF conductivity remains an actual and important task.