Chapter 13 of [[reader_006.pdf]] Chapter 10 of [[incropera.pdf]] ## General Approach 1. Determine regime 2. Apply appropriate correlation 3. Calculate $\mathrm{Nu}$ and $h_\text{eff}$ ## Pool Boiling Plot heat flux against superheat temperature, which is defined as $\Delta T_{e}=T_{s}-T_\text{sat}$ ![[Pasted image 20250418132855.png|425]] ### Natural Convection Boiling - Occurs near saturation temperature - Minimal superheat - Heat transfer via convection - End of this region marked by *onset of nucleate boiling (ONB)* ### Nucleate Boiling - Surface superheat causes bubbles to form at nucleation sites - Jets form at point $B$, $h$ peaks at point $P$ - Very efficient heat transfer - Most systems operate in this regime #### Critical Heat Flux (CHF) Peak $q''$ on the boiling curve $q''_{\text{crit}} = C \cdot h_{fg} \cdot \rho_v \cdot \left( \frac{\sigma g (\rho_l - \rho_v)}{\rho_v^2} \right)^{1/4}$ Typical value: $C \approx 0.131$ for large horizontal cylinders #### Bubbles Bubble diameter is derived from buoyancy force and surface tension $D_{b}\propto \sqrt{ \frac{\sigma}{g(\rho_{l}-\rho_{v})} }$ $\sigma =$ surface tension (N/m) Characteristic velocity is found to by dividing diameter by the time it takes to fill the space behind it: $V \propto \frac{q''_{s}}{\rho_{l}h_{fg}}$ Together with $\overline{Nu}_{L}=C_{fc}R_{L}^{m}Pr^{n}$ we find the following nucleate boiling correlation: $q''_{s}=\mu_{l}h_{fg}\left( \frac{g(\rho_{l}-\rho_{v})}{\sigma} \right)^{1/2}\left( \frac{c_{p,l}\Delta T_{e}}{C_{s,f}h_{fg}\mathrm{Pr}^{n}_{l}} \right)^{3}$ See table for $C_{s,f}$ and $n$ values[^1], but generally $C_{s,f}\approx 1$ and $n=1$ ### Transition Boiling - Unstable regime between nucleate and film boiling - Heat flux drops as vapor patches form ### Film Boiling - Entire surface covered by vapor blanket - Heat transfer drops significantly due to vapor insulation Film boiling on a cylinder or sphere, with $C=0.62$ for horizontal cylinders and $C=0.67$ for sphers $\overline{Nu}_{D} = C \left( \frac{g(\rho_{l}-\rho_{v})h_{fg}'D^{3}}{v_{v}k_{v}(T_{s}-T_{\text{sat}})} \right)^{1/4}$ $h'_{fg}=h_{fg}+0.80c_{p,v}(T_{s}-T_{\text{sat}})$ At surface temperatures greater than $300°C$, radiation is significant: ${} \bar{h}^{4/3}=\bar{h}^{4/3}_{\text{conv}}+\bar{h}_{\text{rad}}\bar{h}^{1/3} {}$ $\bar{h}_\text{rad}=\frac{\varepsilon\sigma(T_{s}^{4}-T_\text{sat}^{4})}{T_{s}-T_\text{sat}}$ $\varepsilon =$ emissivity $\sigma =$ Stefan-Boltzmann constant ## Forced Boiling [^1]: Nucleate boiling correlations ![[Pasted image 20250418143100.png|300]]