## 2 Modeling
### Mathematical Models and Correlations
#### 2.2.2 Mini-Channel Evaporator Model
\begin{array}{l}\text{dT}_{\text{sat,onb}}=\dfrac{4 \cdot\sigma\cdot\text{T}_{\text{sat}}\cdot\left(\text{Sv}_{\text{g}}-\text{ Sv}_{\text{V}1}\right)\cdot\text{h}\alpha_{\text{t}}}{\lambda_{\text{t}} \cdot\left(\text{h}_{\text{g}}-\text{h}_{\text{I}}\right)}\cdot\left[1+\sqrt{1+ \dfrac{\lambda_{1}\cdot\left(\text{h}_{\text{g}}-\text{h}_{\text{I}}\right) \cdot\text{dT}_{\text{sub}}}{2\cdot\sigma\cdot\text{T}_{\text{sat}}\cdot \left(\text{Sv}_{\text{g}}-\text{Sv}_{\text{I}}\right)\cdot\text{h}\alpha_{ \text{I}}}}\right]\\ \text{q}_{\text{onb}}=\left[\dfrac{\lambda_{\text{t}}\cdot\left(\text{h}_{ \text{g}}-\text{h}_{\text{I}}\right)}{8\cdot\sigma\cdot\text{T}_{\text{sat}} \cdot\left(\text{Sv}_{\text{g}}-\text{Sv}_{\text{I}}\right)}\right]\cdot( \text{dT}_{\text{sat,onb}})^{2}\end{array} \right. \tag{15}\]  
Following the ONB is the partial boiling region. In this region, heat transfer is a blend of single-phase convective and nucleate boiling modes. As liquid subcooling decreases, the nucleation activity intensifies along the flow. At a distinct point, the nucleate boiling mode assumes the role of the primary heat transfer mode, with a smaller contribution from the single-phase convective mode. This transition signifies the start of fully developed boiling. Equation (16) determines the heat flux at this juncture (qE):  
\[\left\{ \begin{aligned} & 1058\!\cdot\!\mathrm{F_{fl}}\!\cdot\! \left[\mathrm{Mf}\!\cdot\!\left(\mathrm{h_{g}}-\mathrm{h_{l}}\right)\right]^{-0.7}\!\cdot\!\mathrm{q_{F}}-\mathrm{q_{F}}^{0.3}=1058\!\cdot\!\mathrm{h}\alpha_{ 1}\!\cdot\!\mathrm{F_{fl}}\!\cdot\!\left[\mathrm{Mf}\!\cdot\!\left(\mathrm{h_{ g}}-\mathrm{h_{l}}\right)\right]^{-0.7}\!\cdot\!\mathrm{dT_{sub}}\\ &\mathrm{q_{E}}=1.4\!\cdot\!\mathrm{q_{F}}\end{aligned} \right. \tag{16}\]  
As the degree of supercooling continues to decrease, growing visible bubbles begin to appear in the fluid. The point of net vapor generation denotes the location in the subcooled flow where the net void fraction begins to increase substantially. [PERSON] and [PERSON]'s correlations [42] are employed to locate the thermodynamic quality (xnvg) at the start of significant void flow regions, calculated using Equation (17):  
\[\left\{ \begin{array}{cc}\mathrm{x_{nvg}}=-0.0022\!\cdot\!\mathrm{Bo}\!\cdot \!\mathrm{Re}_{1}\!\cdot\!\mathrm{Pr}_{1}&\left(\mathrm{Re}_{1}\!\cdot\! \mathrm{Pr}_{1}<70,000\right)\\ \mathrm{x_{nvg}}=-154\!\cdot\!\mathrm{Bo}&\left(\mathrm{Re}_{1}\!\cdot\! \mathrm{Pr}_{1}>70,000\right)\end{array} \right. \tag{17}\]  
The heat transfer rates in the partial boiling and fully developed boiling regions can be solved by Equations (18) and (19), respectively:  
\[\left\{ \begin{aligned} &\mathrm{h}\alpha_{\mathrm{pb}}=\frac{ \mathrm{q_{pb}}}{\mathrm{T_{wall}}-\mathrm{T}}\\ &\mathrm{q_{pb}}=\mathrm{a}+\mathrm{b}\!\cdot\!\left(\mathrm{T_{wall }}-\mathrm{T_{sat}}\right)^{\mathrm{m}}\\ &\mathrm{a}=\mathrm{q_{onb}}-\mathrm{b}\cdot\left(\mathrm{dT_{sat, onb}}\right)^{\mathrm{m}}\\ &\mathrm{b}=\frac{\mathrm{q_{E}}-\mathrm{q_{onb}}}{\left(\mathrm{dT_{ sat,E}}\right)^{\mathrm{m}}-\left(\mathrm{dT_{sat,onb}}\right)^{\mathrm{m}}}\\ &\mathrm{m}=\mathrm{n}+\mathrm{p}\!\cdot\!\mathrm{q_{pb}}\\ &\mathrm{n}=1-\mathrm{p}\!\cdot\!\mathrm{q_{onb}}\\ &\mathrm{p}=\frac{\left(\frac{1}{0.3}-1\right)}{\left(\mathrm{q_{E}}- \mathrm{q_{onb}}\right)}\end{aligned} \right. \tag{18}\]  
\[\mathrm{h}\alpha_{\mathrm{fdb}}=\left(1058\!\cdot\!\mathrm{Bo}^{0.7}\!\cdot\! \mathrm{F_{fl}}\!\cdot\!\mathrm{h}\alpha_{1}\right)\!\cdot\!\left(\frac{ \mathrm{T_{wall}}-\mathrm{T_{sat}}}{\mathrm{T_{wall}}-\mathrm{T}}\right) \tag{19}\]  
The heat transfer rate calculations in the significant void flow regions utilize the same equation (Equation (13)) as in saturated flow boiling, substituting the apparent quality x\(\alpha\) for x in Equation (14).