## 2 Heat and Mass Balance Equations for Adsorber of Adsorption Heat Pump
Vapor temperature and pressure during the adsorption process are equal to the evaporator temperature and pressure. 7. Vapor temperature during desorption process is related to the heat balance of the bed, and the vapor pressure during desorption process is equal to the condenser pressure. 8. Switching between adsorption and desorption cycles (cooling water is switched to hot water) takes place in an instant. 9. Desorption kinetics is described and modeled as the adsorption kinetics. The heat and mass balance PDEs were formulated for calculation element \(dx\) (Figure 2). Those equations took into account: heat flow supplied to the process, the heat flow generated, the heat flow discharged from the process, and the heat flow accumulated. In this article, the authors focused on describing methods for solving the set of partial differential equations and other assumptions. A detailed description of the developed model can be found in articles [38; 39], where the performance of two-bed as well as six-bed single-stage adsorption heat pumps were considered. The set of heat and mass balance equations for the adsorber subelement \(dx\) (Figure 2) of the considered adsorption heat pump can be described by  
\[\left\{ \begin{array}{cc}\frac{\partial T_{f}}{\partial t}=A_{f}\Big{(}T_{m} -T_{f}\Big{)}+B_{f}\frac{\partial T_{f}}{\partial x}+C_{f}\frac{\partial^{2} T_{f}}{\partial x^{2}}&(1)\\ \frac{\partial T_{m}}{\partial t}=A_{m}(T_{a}-T_{m})+B_{m}\Big{(}T_{m}-T_{f} \Big{)}+C_{m}\frac{\partial^{2}T_{m}}{\partial x^{2}}\,,&(2)\\ \frac{\partial T_{a}}{\partial t}=A_{a}\frac{\partial a}{\partial t}+B_{a}(T_ {a}-T_{m})+C_{a}\frac{\partial^{2}T_{a}}{\partial x^{2}}&(3)\\ \frac{\partial a}{\partial t}=k_{m}(T_{a})\big{(}a_{eq}(T_{a})-a\big{)}&(4) \end{array} \right. \tag{2}\]  
where constant parameters \(A\), \(B\), and \(C\) from the above set are listed in Table 1. Equation (1) is the fluid (water) energy balance equation, where \(\alpha_{mf}\) is the fluid/metal tube heat transfer coefficient, \(\lambda_{f}\) is the fluid thermal conductivity, and \(R_{1}\) is the tube internal radius. Equation (2) is the heat energy balance for the metal tube, where \(\alpha_{af}\) is the adsorbent/fluid heat transfer coefficient, \(\lambda_{m}\) is the metal tube thermal conductivity, and \(R_{2}\) is the tube outer radius. Equation (3) is the heat and mass balance equation for the adsorbent, where \(\lambda_{a}\) is the adsorbent thermal conductivity and \(\Delta H\) is the isosteric heat of adsorption. Equation (4) is an adsorption and desorption process dynamics described by the application of the linear driving force model (LDF), where \(k_{m}=\frac{15\ D_{e}}{r_{a}^{2}}\) is the mass transport equation and \(D_{e}=D_{0}\ exp\Big{[}-\frac{E_{a}}{R\ T}\Big{]}\)[16]. This equation calculates the time dependent change in concentration (uptake) for the working bed (silica gel) subelement. The heat of adsorption for both the adsorption and desorption processes was determined by the Clausius-Clapeyron equation [33]. All components were assumed at an initial temperature of \(T_{0}\), while initial uptake was considered for initial pressure \(P_{0}\):  
\[\left(T_{f}\right)_{t=0}=\left(T_{m}\right)_{t=0}=\left(T_{a}\right)_{t=0}=T_{0}, \tag{5}\]  
\[\left(a\right)_{t=0}=a_{eq}(P_{0},\ T_{0}). \tag{6}\]  
For adsorption and desorption pressures, it was assumed  
\[P_{ads}=P_{evap}\ \text{and}\ P_{des}=P_{cond}. \tag{7}\]  
The boundary conditions for the fluid, considering the temperature of the process water supplied to the column, were equal to the driving heat temperature:  
\[\left(T_{f}\right)_{x=0}=T_{f\ ads}\ \text{(adsorption)}, \tag{8}\]  
\[\left(T_{f}\right)_{x=0}=T_{f\ des}\ \text{(desorption)}. \tag{9}\]  
The model assumed perfect insulation.