## 2 Modeling
### Mathematical Models and Correlations
#### 2.2.2 Mini-Channel Evaporator Model
The heat transfer process of the refrigerant within the mini-channel evaporators includes single-phase convection, subcooled flow boiling, and saturated flow boiling. 1. Single-phase convection  
Single-phase convection in the mini-channel evaporator is detailed in Equation (13). For the heat transfer correlation of the refrigerant in the single-phase region, the [PERSON] and [PERSON] model [34] is used in the case of laminar flow. For the turbulent heat transfer correlation of the refrigerant in the two-phase region, the Gnielinski model [35], as suggested by [PERSON] [36], is employed. \[\left\{ \begin{array}{l}\text{h}\alpha_{\text{sp}}=\dfrac{\text{Nu}_{ \text{sp}}\cdot\lambda}{\text{D}}\\ \text{Nu}_{\text{sp}}=3.6+\dfrac{0.44\cdot\dfrac{\text{Ew}}{\text{Eh}}}{1+0.56 \cdot\left(\dfrac{\text{Ew}}{\text{Eh}}\right)^{\frac{1}{3}}}\\ \text{Nu}_{\text{sp}}=\dfrac{\left(\dfrac{\text{f}}{2}\right) \cdot\left(\text{Re }-1000\right)\cdot\text{Pr}}{1+12.7\cdot\left(\dfrac{2}{\text{Pr }\,\overline{3}\,-1}\right)\cdot\sqrt{\dfrac{\text{f}}{2}}}\end{array} \right. :\text{Laminarflow}\\ \text{:Turbulentflow} \tag{13}\]  
Here, Ew and Eh denote the width and height of the channel, respectively, while \(\lambda\) symbolizes the thermal conductivity of the refrigerant. Furthermore, Pr refers to the Prandtl number. 2. [wide=0.5 em,leftmargin=0 em]  
The correlation for saturated flow boiling in the mini-channel evaporator is delineated by Equation (14), proposed by [PERSON] [37]:  
\[\left\{ \begin{array}{l}\text{h}\alpha_{\text{tp}}=\text{large}\,\text{rof} \begin{cases}\dfrac{\text{h}\alpha_{\text{tp,nbd}}}{\text{h}\alpha_{\text{tp, cbd}}}&(\text{Re}_{\text{t}}>3000)\\ \text{A linear interpolation between Re}_{\text{t}}\text{ of }1600\text{ and }3000&(1600\leq\text{Re}_{\text{t}}\leq 3000)\\ \text{h}\alpha_{\text{tp}}=\text{h}\alpha_{\text{tp,nbd}}&(\text{Re}_{\text{t} }<1600)\\ \text{h}\alpha_{\text{tp,nbd}}=0.6683\cdot\text{Co}^{-0.2}\cdot(1-\text{x})^{0.8 }\cdot\text{h}\alpha_{\text{t}}+1058.0\cdot\text{Bo}^{0.7}\cdot(1-\text{x})^{0.8 }\cdot\text{F}_{\text{f}}\cdot\text{h}\alpha_{\text{t}}\\ \text{h}\alpha_{\text{tp,cbd}}=1.136\cdot\text{Co}^{-0.9}\cdot(1-\text{x})^{0.8 }\cdot\text{h}\alpha_{\text{t}}+667.2\cdot\text{Bo}^{0.7}\cdot(1-\text{x})^{0.8 }\cdot\text{F}_{\text{f}}\cdot\text{h}\alpha_{\text{t}}\\ \text{Bo}=\dfrac{\text{q}}{\text{G}\cdot\left(\text{h}_{\text{g}}-\text{h}_{ \text{t}}\right)}\\ \text{Co}=\left(\dfrac{\text{\rho}_{\text{g}}}{\text{\rho}_{\text{I}}}\right) ^{0.5}\cdot\left(\dfrac{1-\text{x}}{\text{x}}\right)^{0.8}\end{array} \right. \tag{14}\]  
The calculations of Rel and hacl rely on Equations (5) and (13), respectively, considering the refrigerant in its pure liquid phase. Here, q represents the heat flux, Ffl denotes the fluid-surface parameter, as established by [PERSON]'s studies [38; 39]. 3. [wide=0.5 em,leftmargin=0 em]  
Subcooling flow boiling correlation  
The subcooling flow boiling model, along with the corresponding correlation, builds upon [PERSON]'s research [40; 41]. [PERSON] divided subcooling flow boiling into distinct regions: partial boiling, fully developed boiling, and significant void flow regions. Effective modeling of these regions requires careful consideration of the onset points and distinct heat transfer characteristics of each region. The inception of partial boiling, also known as the onset of nucleate boiling (ONB), signifies the threshold heat flux or the wall superheat at which surface nucleation sites become active, initiating vapor bubble generation. The wall superheat of ONB (dT\({}_{\text{sat,onb}}\)) and heat flux of ONB (q\({}_{\text{onb}}\)) are calculated by Equation (15):  
\[\left\{