## Pure Substance Model A pure substance is a system which is *uniform* and *invariable in chemical composition* Only consider the following: - Energy exchange via heat transfer - Work transfer through $PdV$ Quality factor is the ratio of vapor mass to total saturated mixture mass $x = \frac{m_{g}}{m_{f}+m_{g}}$ If you know any two intensive properties, you can solve for quality factor and everything else (State principle) $x = \frac{v-v_{f}}{v_{fg}}=\frac{u-u_{f}}{u_{fg}}=\frac{h-h_{f}}{h_{fg}}=\frac{s-s_{f}}{s_{fg}}$ ### P-V Diagram ![[P-V-phase-diagram-the-dashed-line-represent-isotherms.png|425]] ### T-V Diagram ![[Pasted image 20250517001827.png|300]] ### P-T Diagram ![[Pasted image 20250407005849.png|525]] ### T-S ![[Pasted image 20250406141948.png|525]] https://demonstrations.wolfram.com/TemperatureEntropyDiagramForWater/ ### H-S ![[Pasted image 20250406141652.png|525]] https://demonstrations.wolfram.com/EnthalpyEntropyDiagramForWater/ ### Ideal Gas ![[PV-Processes.png]] ### Interpolation for dummies $y = y_\text{low} + \underbrace{ \frac{y_\text{high}-y_\text{low}}{x_{\text{high}}-x_\text{low}} }_{ \text{slope} }\cdot (x-x_{\text{low}})$ ## Boiling 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]] ## Condensation Chapter 13 of [[reader_006.pdf]] Chapter 10 of [[incropera.pdf]] ### Dropwise Condensation ### Filmwise Condensation $\dot{m} = \rho_{l}u_{m}b\delta$ $\mathrm{Re}_{\delta}=\frac{4\Gamma}{\mu_{l}} = \frac{4\rho_{l}u_{m}\delta}{\mu_{l}}$ ![[Pasted image 20250418211250.png|275]] ![[Pasted image 20250421163953.png|400]] #### Laminar Condensation Valid for $\mathrm{Re}_{\delta}\leq 30$ For a vertical plate, we can derive the following: $\Gamma(x) = \frac{\dot{m}(x)}{b} = \frac{g\rho_{l}(\rho_{l}-\rho_{v})\delta^{3}}{3\mu_{l}}$ ${} h_{fg}' = h_{fg}+0.68c_{p,l}(T_\text{sat}-T_{s}) = h_{fg}(1+0.68\mathrm{Ja}) {}$ $\bar{h}_{L}=0.943 \left( \frac{ \rho_{l}g(\rho_{l}-\rho_{v})h'_{fg}k^{3}_{l}}{\mu_{l}(T_\text{sat}-T_{s})L} \right)^{1/4}$ $\overline{Nu}_{L}=\frac{\bar{h}_{L}L}{k_{l}}=0.943 \left( \frac{ \rho_{l}g(\rho_{l}-\rho_{v})h'_{fg}L^{3}}{\mu_{l}k_{l}(T_\text{sat}-T_{s})} \right)^{1/4}$ #### Turbulent Condensation