## 2 Physical and Mathematical Models
### Mathematical Model
#### 2.2.2 Energy Balance for Two-Stage Chiller with Heat Recovery
\begin{array}{l}\frac{d}{dt}\{(M_{cm}*Cp_{m}+M_{con}*Cp_{rou})T_{con}\}=- M_{ac}*hf_{fg}*\frac{dx_{abs}}{dt}-M_{ac}*Cp_{m}*(T_{des}-T_{con})*\\ \frac{dx_{abs}}{dt}+M_{con}*Cp_{pw}*(T_{coni}-T_{con})\end{array}  \tag{7}\]  
The left side represents the required sensible heat transfer of the metallic part of the condenser heat exchanger. The first part of the right side represents latent heat of vaporization for methanol. The second term represents sensible heat transfer from desorber to condenser. The last term represents the heat released by cooling water. The outlet temperature of the condenser water is calculated by LMTD:  
\[T_{con}=T_{con}+(T_{coni}-T_{con})*exp\{-(U_{con}*A_{con})/(M_{con}*Cp_{w})\} \tag{8}\]  
* During switching time The energy balance equation for the bed switches from adsorption to desorption is:  
\[ \begin{array}{l}(M_{ac}*Cp_{ac}+M_{ac}*Cp_{m}*x_{des}+M_{Al}*Cp_{Al}+M_{cu}*Cp _{cu})\frac{dT_{abs}}{dt}\\ =Mc_{w}*Cp_{wu}*(T_{hi}-T_{ho})\end{array}  \tag{9}\]  
The energy balance equation for the bed switches from desorption to adsorption is:  
\[ \begin{array}{l}(M_{ac}*Cp_{ac}+M_{ac}*Cp_{m}*x_{ads}+M_{Al}*Cp_{Al}+M_{cu}*Cp _{cu})\frac{dT_{abs}}{dt}\\ =Mc_{w}*Cp_{wu}*(T_{ci}-T_{co})\end{array}  \tag{10}\]  
For the evaporator:  
\[ \begin{array}{l}(M_{con}(t)*Cp_{m}+M_{cu}*Cp_{rou})\frac{dT_{rou}}{dt}=-M_{ ac}*hf_{fg}*\frac{dx_{abs}}{dt}+M_{chw}*Cp_{pw}*(T_{chi}-T_{chw}).\end{array}  \tag{11}\]  
The outlet temperature of the chilled water is calculated by LMTD:  
\[T_{cho}=T_{eva}+(T_{chi}-T_{eu})*exp\{-(U_{cwu}*A_{eva})/(M_{chw}*Cp_{w})\ \} \tag{12}\]  
For the condenser:  
\[ \begin{array}{l}(M_{cm}*Cp_{m}+M_{con}*Cp_{rou})\frac{dT_{rou}}{dt}=-M_{ ac}*hf_{fg}*\frac{dx_{abs}}{dt}+M_{con}*Cp_{pw}*(T_{coni}-T_{con}).\end{array}  \tag{13}\]The outlet temperature of the condenser water is calculated by Log Mean Temperature Difference (LMTD):  
\[T_{cono}=T_{con}+(T_{coni}-T_{con})*exp\{-(U_{con}*A_{con})/(M_{con}*Cp_{w})\} \tag{14}\]  
C. Heat Recovery:  
G1 is connected to G3: the hot water outlet of G3 is recirculated through G1. G2 is connected to G4: the hot water outlet of G2 is recirculated through G4. No external heating or cooling. For G1 and G2:  
\[ \begin{array}{c}(M_{ac}*Cp_{ac}+M_{Al}*Cp_{Al}+M_{cu}*Cp_{cu})\frac{dT_{ dis}}{dt}\\ =M_{ac}*Q_{st}*\frac{dx_{dis}}{dt}-M_{ac}*Cp_{m}*T_{ds}*\frac{dx_{dis}}{dt}+Mre_{ w}*Cp_{w}*(T_{ho3}-T_{co1})\end{array}  \tag{15}\]  
For G3 and G4:  
\[ \begin{array}{c}(M_{ac}*Cp_{ac}+M_{Al}*Cp_{Al}+M_{cu}*Cp_{cu})\frac{dT_{ dis}}{dt}\\ =M_{ac}*Q_{st}*\frac{dx_{dis}}{dt}-M_{ac}*Cp_{m}*T_{ds}*\frac{dx_{dis}}{dt}+Mre_{ w}*Cp_{w}*(T_{co1}-T_{ho3})\end{array}  \tag{16}\]  
For the evaporator:  
\[ \begin{array}{c}(M_{em}(t)*Cp_{m}+M_{eva}*Cp_{eva})\frac{dT_{ com}}{dt}=-M_{ac}*h_{fg}*\frac{dx_{dis}}{dt}-M_{ac}*Cp_{m}*\\ (T_{con}-T_{eva})*\frac{dx_{dis}}{dt}+M_{clw}*C_{pw}*(T_{chi}-T_{cho})\end{array}  \tag{17}\]  
The outlet temperature of the chilled water is calculated by LMTD:  
\[T_{cho}=T_{eva}+(T_{chi}-T_{eva})*exp\{-(U_{cna}*A_{eva})/(M_{chw}*Cp_{w})\ \} \tag{18}\]  
For the condenser:  
\[ \begin{array}{c}(M_{cm}*Cp_{m}+M_{con}*Cp_{con})\frac{dT_{ com}}{dt}=-M_{ac}*h_{fg}*\frac{dx_{dis}}{dt}-M_{ac}*Cp_{m}*(T_{ds}-T_{con})*\\ \frac{dx_{dis}}{dt}+M_{clw}*Cp_{w}*(T_{coni}-T_{com})\end{array}