## 3 Estimation of Storage Performance
#### Determination of the Storage Energy Efficiency  
Storage efficiency is a parameter calculated to compare different TES systems. In its most classical definition, storage energy efficiency can be described as the ratio between the useful energy and the input energy of the storage system [21]:\[\eta_{storage}=\frac{Energy_{useful}}{Energy_{charge}} \tag{5}\]  
In this paper, the storage efficiency is calculated considering the best case, in which it is possible to recover both the heat exiting the condenser (for heating operations) and the heat required by the evaporator (for cooling operations). The efficiency evaluation refers to two fundamental conditions. The first one is for short-term storage (ideally with zero storage time); the efficiency calculation is described by the following formula:  
\[\eta_{storage\ (t=0)}=\frac{Q_{3}+Q_{4}+Q_{cond}+Q_{cusp}}{Q_{1}+Q_{2}} \tag{6}\]  
The second condition represents long-term (ideally infinite) storage. In this case, the stored sensible heat related to the four phases of the thermodynamic cycle is totally dissipated. The useful heat is therefore reduced to the heat associated with the condenser and the evaporator, in addition to the adsorption heat (\(Q_{h}\)):  
\[\eta_{storage\ (t=\infty)}=\frac{Q_{h}+Q_{cond}+Q_{cusp}}{Q_{1}+Q_{2}} \tag{7}\]  
The calculation of heat flows follows the indications given in Ref. [22]. The heat of the condenser and evaporator can be calculated as a function of temperature:  
\[Q_{*}(T)=Q_{*}^{\prime}(T)+Q_{sv}(T) \tag{8}\]  
\(Q_{*}^{\prime}\) is the latent heat of the refrigerant at temperature \(T\), relative to the amount of refrigerant desorbed:  
\[Q_{*}^{\prime}(T)=\Delta wL(T) \tag{9}\]  
\(Q_{sv}\) refers to the sensible heat released (or supplied) to cool (or heat) the vapour until it reaches the condensation (or adsorption) temperature:  
\[Q_{sv}(T)=C_{p_{v}}(T)\ \Delta w\ (T_{avg}-T) \tag{10}\]  
where \(C_{p_{v}}(T)\) represent the specific heat of the refrigerant vapour and the temperature \(T_{avg}\) is defined as:  
\[T_{avg}= \begin{cases}\frac{T_{1}+T_{4}}{2}&for\ T=T_{cusp}\\ \frac{T_{2}+T_{3}}{2}&for\ T=T_{cond}\end{cases}  \tag{11}\]  
The term \(Q_{i}\), for \(i=1\)-4, represents the total heat associated with each of the four phases that make up the storage cycle, and is made up of three contributions:  
\[Q_{i}=Q_{s,i}+\beta Q_{h}-\gamma Q_{sv}(T_{cusp}) \tag{12}\]  
with \(\beta=0\) for the isosteric heating and cooling phases (\(i=1,3\)) and \(\beta=1\) for the isobaric desorption and adsorption phases (\(i=2,4\)); \(\gamma\) is equal to zero for i = 1, 2, 3 while for Phase 4 (isobaric adsorption) it is \(\gamma=1\). \(Q_{s,i}\) represents the sensible heat associated with heating or cooling the adsorbent bed. It consists of a term relating to the adsorbent rich in adsorbed refrigerant and a term relating to the metal mass present in the bed (\(\phi\) represents the metal/adsorbent mass ratio):  
\[Q_{s,i}=\int\limits_{T_{i}}^{T_{i+1}}\Big{[}C_{p_{q}}(w(T),T)+C_{p_{out}}\phi \Big{]}dT \tag{13}\]where \(w(T)\) is equal to \(w_{max}\) for \(i=1\) and it is equal to \(w_{min}\) for \(i=3\), while for \(i=2,4\) (isobaric phases of desorption and adsorption) an average value between \(w_{min}\) and \(w_{max}\) is considered. The equivalent specific heat is a function of the uptake and the temperature and can be calculated as:  
\[C_{p_{eq}}(w,T)=C(w)+D(w)(T-273.15) \tag{14}\]  
where \(C(w)\) and \(D(w)\) are second-order polynomials described by experimental coefficients specific for each working pair:  
\[C(w)=c_{0}+c_{1}w+c_{2}w^{2} \tag{15a}\] \[D(w)=d_{0}+d_{1}w+d_{2}w^{2} \tag{15b}\]  
The experimental coefficients \(c_{i}\) and \(d_{i}\) are not available for all the considered working pairs.