## 3 Estimation of Storage Performance  
The purpose of the study is the estimation of the storage performance for a preliminary evaluation of the potentialities of the proposed system and for a comparison of different working pairs. The evaluation is carried out through a simplified and robust mathematical model. The model used for the study has the objective of describing a thermodynamic cycle of a storage unit, usually represented in a Clapeyron diagram (see Figure 4), to determine the corresponding efficiency and allow the comparison of different adsorbent materials.  
As already introduced in the previous sections, the basic storage sorption cycle consists of four phases grouped in the two typical stages of a storage system: the charging phase  
Figure 4: Thermodynamic cycle of an adsorption storage unit.  
consisting of isosteric heating (Phase 1), isobaric desorption (Phase 2), the discharging phase consisting of an isosteric cooling (Phase 3), and isobaric adsorption (Phase 4).  
To facilitate the study, the following assumptions were made:  
* the temperatures of the evaporator and condenser are constant and are indicated as \(T_{eup}\) and \(T_{cend}\);
* the minimum and maximum temperatures of the cycle are known: \(T_{min}\equiv T_{1}\) and \(T_{max}\equiv T_{3}\);  
Once \(T_{eup}\) and \(T_{cend}\) are known, the pressure of the evaporator and condenser (\(P_{eup}\) and \(P_{cend}\)) can be determined. The equilibrium of adsorbent and adsorbate can be described by several models reported in literature (e.g., [PERSON]'s model, Dubinin-Radushkevich, etc.); however, in this study, the Clausius-Clapeyron phenomenological law was implemented [20], as it revealed to be adequate to describe the adsorbent/adsorbate equilibrium for the more promising working pairs identified for application in a storage unit.  
An iterative procedure was used to determine the uptake: the maximum and minimum uptake values, which correspond to the two isosteric conditions of the thermodynamic cycle, are established based on the thermal-pressure conditions. The Newton-Raphson method is used to calculate the uptake from the non-linear equation describing the equilibrium between adsorbent and adsorbate:  
\[lnP=A(w)+\frac{B(w)}{T} \tag{1}\]  
where \(A(w)\) and \(B(w)\) are third-order polynomials, calculated through coefficients determined experimentally and which depend on the working pair:  
\[A(w) =a_{0}+a_{1}w+a_{2}w^{2}+a_{3}w^{3} \tag{2a}\] \[B(w) =b_{0}+b_{1}w+b_{2}w^{2}+b_{3}w^{3} \tag{2b}\]  
The temperatures \(T_{2}\) and \(T_{4}\), corresponding, respectively, to the start of Phase 2 and the start of Phase 4 of the cycle, can be determined as a function of the operating conditions:  
\[T_{2} =T(P_{cend},w_{max}) \tag{3a}\] \[T_{4} =T(P_{eup},w_{min}) \tag{3b}\]  
The mentioned calculations allow the determination of the three values (pressure, temperature and uptake) that allow the identification of the four points that define the thermodynamic cycle of the storage system.  
\[\left\{\begin{aligned} P_{eup},&\;T_{1}\equiv T_{ min},w_{max}\\ P_{cend},&\;T_{2},w_{max}\\ P_{cend},&\;T_{3}\equiv T_{max},w_{min}\\ P_{eup},&\;T_{4},w_{min}\\ \end{aligned}\right. \tag{4}\]  
Knowledge of these values allows the calculation of storage efficiency.