## 2 Materials and Methods
### Methods  
#### 2.2.1 VPD Control Development  
Mathematically, VPD is defined as the difference between the saturation vapor pressure \((VPD_{sat})\) related to temperature and the air vapor pressure \((VPD_{air})\) associated with the relative humidity. Equations (1)-(3) presented by [36] and used in an investigation by [33] were used to calculate the VPD. \[VPD=[VP_{sat}-VP_{air}] \tag{1}\]  
\[VP_{sat}=0.61078\bigg{(}e^{[[\frac{[27,27](7)}{29,37+1}]}\bigg{)} \tag{2}\]  
\[VP_{air}=\bigg{(}\frac{RH}{100}\bigg{)}(VP_{sat}) \tag{3}\]  
Figure 1: Hood-type micro-greenhouse prototype. Figure 2: Components (**a**) and dimensions (mm) (**b**) of the micro-greenhouse. Temperature  
For this research, an unmixed crossflow heat exchanger was used for temperature control, as it is a heat exchanger in which two different fluids can interact, in this case, water and air. Heat exchangers regularly operate for long periods of time, without significant changes in their operating conditions. For that reason, the analysis can be considered as a steady flow system. The heat exchanger's actual heat transfer rate can be defined based on an energy balance in the hot and cold liquids [37]. For the mathematical analysis of the crossflow heat exchanger for heating the temperature system, in one exchanger, one fluid gives up energy and the other acquires energy; therefore, the heat transfer that exists in the water is equal to the heat transfer in the air, as shown in Equations (4)-(6). \[\dot{Q_{w}}=\dot{Q_{a}} \tag{4}\]  
\[\dot{Q_{w}}=\dot{m_{w}}c_{pw}(T_{w,in}-T_{w,out}) \tag{5}\]  
\[\dot{Q_{a}}=\dot{m_{a}}c_{pa}(T_{a,out}-T_{a,in}) \tag{6}\]  
where \(\dot{Q_{w}}\) = current heat transfer from water, \(\dot{Q_{a}}\) = current heat transfer from air, \(\dot{m_{w}}\) = water mass flow, \(m_{a}\) = air mass flow, \(c_{pw}\) = specific water heat, \(c_{pa}\) = specific air heat, \(T_{w,in}\) = water input temperature, \(T_{a,in}\) = air input temperature, \(T_{w,out}\) = water output temperature and \(T_{a,out}\) = air output temperature. The \(T_{a,out}\) is the internal micro-greenhouse temperature; therefore, Equation (7) is obtained for heating. \[\frac{\dot{m_{w}}c_{pw}(T_{w,in}-T_{w,out})+\dot{m_{a}}c_{pa}T_{a,in}}{\dot{m_ {a}}c_{pa}}=T_{a,out} \tag{7}\]  
Figure 3: Schematic of heat exchangers for heating (**a**) and cooling (**b**). The variable \(T_{a,out}\) represents the output temperature of the air when it passes through the heat exchanger. The mathematical analysis for the cooling cross flow heat exchanger in the temperature system is similar to the heating system, except that the signs are reversed because in heating, the \(T_{a,out}\) gains energy, and in cooling it loses energy. That is, when the system increases its temperature, it absorbs heat, therefore it is said that the system gains energy. The principle remains that the water heat transfer is equal to that of air, as shown in Equation (8). \[VP_{sat}=0.61078\bigg{(}e^{[[\frac{[27,27](7)}{29,37+1}]}\bigg{)} \tag{2}\]0  
Equations (7) and (8) describe heating and cooling dynamics in the heat exchanger for the temperature system in the micro-greenhouse. Relative Humidity  
For the humidification system, based on the work presented by [PERSON] and [PERSON] [37], the adiabatic saturation process is taken as the basis, in which there is an unsaturated airflow that has a specific initial humidity and an input temperature, gaining humidity when it comes into contact with a water tank, in which the water vapor mixes with the airflow. The humidity content increases during this process. Water temperature is kept at the same temperature as the input air; therefore, there is a similar temperature between the inlet and the outlet. By supplying make-up water to the tank at the evaporation rate and the temperature \(T_{2}\), the process can be analyzed as a steady flow process.