## 2 Thermodynamic Analysis  
In this section, the basic architecture of the AdHT and the thermodynamic analysis to evaluate efficiency and energy density are described. Figure 1 represents the basic architecture of an AdHT. Four main components define the overall operation of an AdHT:  
1. An adsorbent bed, which acts either as adsorber or desorber depending on the working phase of the AdHT cycle. It usually consists of a heat exchanger (HEX) filled with the adsorbent material, to achieve high heat and mass transfer rates during the system operation. 2. A condenser, represented by a HEX in which the refrigerant released by the desorber in the vapor phase is condensed to the liquid phase, releasing the heat to the low temperature sink. 3. An evaporator, represented by a HEX in which the waste heat is supplied to evaporate the refrigerant, which is then adsorbed in the adsorber releasing the upgraded heat. 4. A circulating pump needed to move the liquid refrigerant from the condenser to the evaporator. Figure 2 represents the working cycle of the AdHT on a pressure-temperature chart, plotted over an isosteric diagram of a reference working pair. Specifically, the two limiting isosters, namely the rich isoster (w\({}_{1}\)) and the weak isoster (w\({}_{2}\)), which define the operating range of the AdHT, are reported. The main difference between the AdHT technology and standard cycle (i.e., heat pump or chiller), as already detailed in [20], is that the adsorption (SP) and desorption (QR) processes are pressure-driven isothermal processes, instead of the typical isobaric temperature-driven ad-desorption stages of adsorption heat pumps and similarly to the discharge stage of a sorption seasonal storage [21]. Furthermore, the working fluid is evaporated by exploiting the heat of evaporation supplied by waste heat at a temperature level, T\({}_{\text{L}}\) in Figures 1 and 2, which is higher than the temperature, T\({}_{0}\), at which the condensation heat is dumped to the environment. Accordingly, the pressure inside the condenser is lower than the pressure inside the evaporator, thus making the use of a pump necessary to increase the pressure level of the liquid refrigerant and guarantee the correct operation and refrigerant transfer from the condenser to the evaporator. Figure 1: Schematic architecture of an adsorption heat transformers (AdHT) reporting the main components, the energy fluxes, and the temperature levels. T\({}_{L}\), T\({}_{A}\), and T\({}_{0}\) represent the waste heat temperature, upgraded temperature, and ambient temperature, respectively. The same temperatures marked with the apex considers a possible temperature gradient to drive the heat transfer process. Q\({}_{L}\) and Q\({}^{\prime}\)\({}_{L}\) represent the heat flux of the waste heat to the evaporator and to the adsorbent bed, respectively, while Q\({}_{A}\) and Q\({}_{0}\) represent the heat flux of the upgraded heat and discharged at the condenser, respectively. Figure 2: Thermodynamic cycle of an AdHT plotted over a reference isosteric chart, showing the typical pressure-driven working cycle, highlighting the rich isoster (w\({}_{1}\)) and the weak isoster (w\({}_{2}\)). P\({}_{0}\) and P\({}_{L}\) represent the pressure corresponding to the ambient temperature and waste heat temperature levels respectively. T\({}_{L}\), T\({}_{A}\), and T\({}_{0}\) represent the waste heat temperature, upgraded temperature, and ambient temperature, respectively. Q\({}_{L}\) and Q\({}^{\prime}\)\({}_{L}\) represent the heat flux of the waste heat to the evaporator and to the adsorbent bed, respectively, while Q\({}_{A}\) and Q\({}_{0}\) represent the heat flux of the upgraded heat and discharged at the condenser, respectively. The AdHT cycle can be described following the representation reported in Figure 2. It mainly consists of two phases: A pre-cooling and desorption phase (P-Q-R) and a pre-heating and adsorption phase (R-S-P).