## 3 Mathematical Modeling
### Equilibrium Analysis of the Adsorption Heat Pump Cycle  
Many studies have already adopted the idea of the adsorption phenomenon into heat pump systems. [PERSON] [51] presented the concept of optimal adsorbents' parameters for adsorptive cooling/heating applications focusing on the limitations and potential of the recent materials, and [PERSON] et al. [52] reviewed the current situation of the heating applications that are using adsorption heat pumps. Here, it is necessary to point out that most of the research so far has focused mostly on cooling applications. However, heating applications are crucial for sustainable living conditions as well and will be an essential part of future energy transformations. Therefore, in this study, we focus on the heating part of the adsorption heat pump system. The cycle schematics of the adsorption heat pump system are shown in the diagram in Figure 4.  
Formerly, [PERSON] et al. [53] expressed the detailed thermodynamic processes of the ideal adsorption cycle, and this ideal approach of a single bed adsorption cycle configuration is also shown for our intended system in Figure 4 by the solid line. It consists of four sequential thermodynamic processes: I. Pre-heating (b-c), II. Desorption (c-d), III. Pre-cooling (d-a), and IV. Adsorption (a-b). \(Q_{input}\) is the heat supply to the adsorption bed heat exchanger, and it is used for heating the bed during the desorption process. \(Q_{cond}\) is the heat released in the condenser, \(Q_{out}\) is the heat produced during the adsorption process extracted from the bed, and \(Q_{cut}\) is the heat supply to the evaporator during evaporation of the adsorbate. Heat input of the cycle is the same as for the cooling mode; however, in the heating mode, instead of specific cooling power (SCP) (kJ/kg), specific heating power (SHP) (kJ/kg) is used for the calculation. The following Equations show the calculation of the SHP:  
\[\text{SHP }=\ Q_{cond}+Q_{ads} \tag{27}\]  
\[Q_{cond}=\Big{(}q_{fg}(T_{cond})\Big{)}(w_{max}-w_{min}) \tag{28}\]  
\[Q_{ads}=q_{st}(w_{max}-w_{min}) \tag{29}\]  
The coefficient of performance (COP) of heating mode can be then express as:  
\[\text{COP}_{\text{Eq,heating}}=\ \frac{SHP}{Q_{IN}} \tag{30}\]  
where \(Q_{IN}\) is the total heat input of the adsorption system. \(Q_{IN}\) consists of two parts: latent heat and sensible heat. Equations (31)-(33) show the necessary calculation for the heating mode COP:  
\[Q_{IN}=Q_{Lat}+Q_{Sen} \tag{31}\]  
\[Q_{Lat}=\ q_{st}(T_{des},P_{cond})\int_{T_{Pb},P_{cond}}^{T_{des},P_{cond}}dw \tag{32}\]  
\[Q_{Sen}=\frac{c_{rank}(T_{des})+c_{rank}(T_{des})}{2\int_{T_{ds}}^{T_{des}}dT} +\ w_{max}\Bigg{[}\frac{c_{pa}(T_{Pb},P_{cond})+c_{pa}(T_{ab},P_{ens})}{2\int_{ T_{ps}}^{T_{Pb}}dT}\Bigg{]} \tag{33}\]  
The latent heat \(Q_{Lat}\) in Equation (32) is simply calculated as a function of the adsorption heat \(q_{st}\) in the working range of the desorption temperature \(T_{des}\) and pre-heating temperature \(T_{Pb}\) under the given pressure conditions \(P_{cond}\). The sensible heat \(Q_{Sen}\) is then calculated as the addition of heat contained by all major parts in the working temperature range. Given the unity factor of the specific heat of the adsorber \(c_{p,ads}\) used in the gener  
Figure 4: Schematics of the adsorption heat pump cycle. alized calculation, the heat contained by the heat exchanger can be expressed directly as the ratio to the adsorbed amount \(M_{HX,Bal}\)/\(M_{ads}\) combined with the specific heat of the adsorption bed material. Finally, the amount of previously adsorbed refrigerant is accounted for as well through its specific heat \(c_{p,\mu}\) and specific adsorption amount function \(w\)(\(T\),\(P\)), which is based on the adsorption isotherm modeling. [PERSON] et al. [21] presented their model with the specific heat capacity of the adsorbent as a temperature-dependent variable. Hence, in this work, we also use a similar approach and introduce the specific heat as a function of temperature \(c_{p,abs}\)(\(T\)).