Question 10.17: KNOWN: Saturated water boiling on a brass plate maintained a......
KNOWN: Saturated water boiling on a brass plate maintained at 115°C.
FIND: Power required (W/m²) for pressures of 1 and 10 atm; fraction of critical heat flux at which plate is operating.
ASSUMPTIONS: (1) Nucleate pool boiling, (2) Δ\mathrm T_{\mathrm e} = 15°C for both pressure levels.
PROPERTIES: Table A-6, Saturated water, liquid (1 atm, \mathrm T_{\text{sat}} = 100°C): \rho_{\ell} = 957.9 kg/m³, \mathrm c_{\mathrm p,\ell} = 4217 J/kg·K, \mu_{\ell} = 279 × 10^{-6} N·s/m², Pr = 1.76, \mathrm h_{\mathrm{fg}} = 2257 kJ/kg, σ = 58.9 × 10^{-3} N/m; Table A-6, Saturated water, vapor (1 atm): \rho_{\mathrm v} = 0.596 kg/m³; Table A-6, Saturated water, liquid (10 atm = 10.133 bar, \mathrm T_{\text{sat}} = 453.4 K = 180.4°C): \rho_{\ell} = 886.7 kg/m³, \mathrm c_{\mathrm p,\ell} = 4410 J/kg·K, \mu_{\ell} = 149 × 10^{-6} N·s/m², \mathrm{Pr}_{\ell} = 0.98, \mathrm h_{\mathrm{fg}} = 2012 kJ/kg, σ = 42.2 × 10^{-3} N/m; Table A-6, Water, vapor (10.133 bar): \rho_{\mathrm v} = 5.155 kg/m³.
SCHEMATIC:
ANALYSIS: With \Delta\mathrm T_{\mathrm e} = 15°C, we expect nucleate pool boiling. The Rohsenow correlation with \mathrm C_{\mathrm{s,f}} = 0.006 and n = 1.0 for the brass-water combination gives
q _{ s }^{\prime \prime}=\mu_{\ell} h _{ fg }\left[\frac{ g \left(\rho_{\ell}~-~\rho_{ v }\right)}{\sigma}\right]^{1 / 2}\left\lgroup\frac{ c _{ p , \ell} \Delta T _{ e }}{ C _{ s , f } h _{ fg } \operatorname{Pr}_{\ell}^{ n }}\right\rgroup^3
1 atm: q _{ s }^{\prime \prime} = 279 \times 10^{-6} N \cdot s / m ^2 \times 2257 \times 10^3 J / kg \left[\frac{9.8 m / s ^2(957.9 ~ -~ 0.596) kg / m ^3}{58.9 ~\times ~10^{-3} N / m }\right]^{1 / 2} \times \left\lgroup \frac{4217 J / kg \cdot K ~\times ~15 K }{0.006~ \times ~2257 ~\times ~10^3 J / kg~ \times ~1.76^1}\right)^3=4.70 MW / m ^2
10 atm: q _{ s }^{\prime \prime} = 23.8 MW/m^2
From Example 10.1, \mathrm q_{\max}^{\prime\prime} (1 atm) = 1.26 MW/m². To find the critical heat flux at 10 atm, use the correlation of Eq. 10.7,
q _{\text {max }}^{\prime \prime}=0.149 h _{ fg } \rho_{ v }\left[\sigma g \left(\rho_{\ell} – \rho_{ v }\right) / \rho_{ v }^2\right]^{1 / 4}.
q _{\max }^{\prime \prime}(10 atm )=0.149 \times 2012 \times 10^3 J / kg \times 5.155 kg / m ^3 \times \left[\frac{42.2 ~\times~ 10^{-3} N / m ~\times ~9.8 m / s ^2(886.7~ – ~ 5.16) kg / m ^3}{\left(5.155 kg / m ^3\right)^2}\right]^{1 / 4}=2.97 MW / m ^2.
For both conditions, the Rohsenow correlation predicts a heat flux that exceeds the maximum heat flux, \mathrm q_{\max}^{\prime\prime}. We conclude that the boiling condition with \Delta\mathrm T_{\mathrm e} = 15°C for the brass-water combination is beyond the inflection point (P, see Fig. 10.4) where the boiling heat flux is no longer proportional to \Delta\mathrm T^3_{\mathrm e}.
q _{ s }^{\prime \prime} \approx q _{\max }^{\prime \prime}(1 atm ) \leq 1.26 MW / m ^2 ~~~\quad~~~ q _{ s }^{\prime \prime} \approx q _{\max }^{\prime \prime}(10 atm ) \leq 2.97 MW / m ^2.