# Performance Analysis of the Multi-Stage Dew-Point Indirect Evaporative Air Cooler
## 2 Materials and Methods  
Numerical simulations are performed with mathematical models based on the convective heat transfer coefficient. The general assumptions regarding this modeling method are presented in [18]. Generally, the heat exchange with the surroundings is negligible, the operation is steady-state, air flow is an incompressible gas, the water used for evaporation keeps the plate surface in a saturated state, the mass-flow force follows from the humidity ratio gradient, kinetic properties of air stream and water are constant and assumed equal, and the bulk average value of air-flow heat capacity is much larger than that of water. Model validation is presented in Appendix A. The unit studied can be divided into two main heat and mass transfer zones: the cross-flow zone and the counter-flow zone. In these two areas, the process of heat transfer needs to be described with different sets of equations: ordinary and partial. The sensible heat balance for the primary air flow for the regenerative and cross-flow parts [18] are described as:  
\[\dot{G}_{1}\cdot c_{p1}\cdot\frac{\partial t_{1}}{\partial X}dX=-dQ_{1}^{s}, \tag{1}\]  
where:  
\(\dot{G}_{1}\) --partial air mass flow rate: \(\dot{G}_{1}=\left(\frac{dY}{L_{Y}}\right)\cdot G_{1}\), kg/s  
\(c_{p}\) --specific heat capacity of moist air, J/(kg K)  
\(t\) --temperature, \({}^{\circ}\)C  
\(dQ_{1}^{s}\)--sensible heat transfer rate, W  
\(L_{Y}\)--channel length in \(Y\) direction, m  
\(X\)--coordinate along the supply air-flow direction, m  
Regarding the regenerative part [18]:  
Figure 1: Evaporative coolers schemes. (**a**) Multi-stage dew-point evaporative cooler. (**b**) Regenerative cooler [17]. * The sensible heat balance for the working air flow: \[-\dot{G}_{2}\cdot c_{p2}\cdot\frac{\partial t_{2}}{\partial X}dX=-dQ_{2}^{s}+c_{ g2}\cdot(t^{\prime}{}_{p2}-t_{2})\cdot dM_{2},\] (2)
* The sensible mass balance for the working air flow: \[-\dot{G}_{2}\cdot\frac{\partial x_{2}}{\partial X}dX=dM_{2},\] (3)
* The total heat balance: \[dQ_{1}^{s}-dQ_{2}^{l}-dQ_{2}^{s}=0,\] (4) where: \[\dot{G}_{2}\] --partial moist air mass flow rate: \[\dot{G}_{2}=\left(\frac{dV}{L_{Y}}\right)\cdot G_{2}\], kg/s \[dM\] --mass transfer rate of water vapor, kg/s \[t^{\prime}{}_{p2}\] --plate temperature, \[{}^{\circ}\]C Figure 1:0 --latent heat transfer rate, W Regarding the cross-flow zone [18]:
* The sensible heat balance for the working air flow: Figure 1:1 (5) where: Figure 1:2 --partial moist air mass flow rate: Figure 1:3, kg/s Figure 1:4 --channel length in Figure 1:5 direction, m Figure 1:6 --coordinate along the working air-flow direction, m
* The mass balance for the working air flow: Figure 1:7 (6)
* The total heat balance: Figure 1:8 (7)  
The boundary conditions are presented below on the base of Figure 1:9 Air-flow parameters at the entrance to the dry channels in the regenerative part:  
\[ \begin{array}{c|c}t_{1}&=t_{1i}&x_{1}\\ \overline{X}=0.0&\overline{X}=0.0\\ \overline{Y}=(0.0\dots 1.0)&\overline{Y}=(0.0\dots 1.0)\end{array} \]  
Air-flow parameters at the entrance to the working channels in the regenerative part:  
\[ \begin{array}{c|c}t_{2}&=t_{2i}=t_{1o}&x_{2i}=x_{2i}=x_{1o}=x_{1i}\\ \overline{X}=1.0&\overline{X}=1.0\\ \overline{Y}=(0.0\dots 1.0)&\overline{Y}=(0.0\dots 1.0)\end{array} \]  
Air-flow parameters at the entrance to the working channels in the cross-flow part:  
\[ \begin{array}{c|c}t_{3}&=t_{3i}&x_{2}&=x_{3i}\\ \overline{X}=(0.0\dots X_{cross})&\overline{X}=(0.0\dots X_{cross})\\ \overline{Y}=0.0&\overline{Y}=0.0\end{array} \]The model also includes an empirical equation that describes the water vapor saturation pressure as a function of temperature [19]. Numerical models of typical regenerative air coolers (DPIEC 3) are identical to the model of the regenerative zone of the combined cross-regenerative cooler. The equations presented above were solved using the developed algorithm that was implemented in a digital computer program.