## 2 Materials and Methods
### Cross-Flow Heat and Mass Exchanger  
To evaluate the performance of cross-flow exchanger for different conditions the one-dimensional mathematical model is used. The model considers different operation variants. The general variants are \"dry\", \"wet\", and \"frost\". The model includes a possibility for mixed variants to occur, for example \"dry\" and \"wet\" occurring simultaneously. Those are described in detail in other publications [24; 35]. In this work, the most desirable variant of the exchanger operation is the wet variant, which enables the water recovery from the exhaust air. The operating variants directly depend on the parameters of the exhausted air and outdoor air temperature, hence the mathematical model includes the equations in case of sensible heat transfer in outdoor and exhausted channel (\"dry variant\"), water vapor condensation in the form of water film in exhausted channel (\"wet variant\") and water vapor condensation in the form of ice layer in exhausted channel (\"frost variant\").  
In this study, the cross-flow exchanger model, based on the \(\varepsilon\)-NTU, method was used, detailed assumptions for the model can be found [35]. The equations of the mathematical model were implemented into the original compute program in MATLAB environment. The basic equations of the model are shown below.  
Overall energy balance within the differential element: in the case of \"dry\" variant of heat exchange on the surface of the exhaust air passage occurring when plate temperature is higher than the dew point temperature \((t_{\text{DP}})\) of the exhausted air \(\left(t^{\prime}{}_{p2}=t_{p2}>t_{2}^{DP}\right)\):  
\[\dot{Q}_{1}^{S}=\dot{Q}_{2}^{S}=\dot{Q}_{plt}^{cond} \tag{7}\]  
Overall energy balance within the differential element in the case of \"wet\" variant of heat and mass exchange on the surface of the exhaust air passage occurs when local plate temperature is lower than the dew point temperature of the exhausted air \(\left(t^{\prime}{}_{p2}=t_{p2}<t_{2}^{DP}\right)\) and in case of \"frost\" variant occurring when local plate temperature is lower than \(0^{\circ}\text{C}\left(t^{\prime}{}_{p2}=t_{p2}<0\ ^{\circ}\text{C}\right)\):  
\[\dot{Q}_{1}^{S}=\dot{Q}_{2}^{S}+\dot{Q}_{2}^{L}=\dot{Q}_{plt}^{cond} \tag{8}\]  
Using the mathematical model implemented in Matlab, a black box model was developed to determine the operating parameters of the cross-flow unit. The threshold outdoor air temperature was established by performing a calculation loop for different exhausted air parameters and specific heat exchanger number of heat transfer units (NTU = 8).  
\[t_{thrsld}=f(NTU,V_{2}/V_{1},t_{2i},x_{2i}) \tag{9}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Sensible Heat, kW** & **Latent Heat, kW** \\ \hline Plants & \(-9.9\) & \(29.2\) \\ LED lights & \(66.8\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: The values of calculated the sensible and latent heat loads.  
Since indoor conditions are constant the threshold air temperature is stable and equal to \(-2.5\,^{\circ}\)C. The amount of recovered water was calculated using the equation (where \(i\)--inlet conditions, \(o\)--outlet conditions, and \(\rho_{w}\)--water density (kg/m\({}^{3}\))).  
\[w=\frac{V\rho_{air}(x_{2i}-x_{2o})}{\rho_{w}} \tag{10}\]