Question 10.24: KNOWN: Operating conditions of apparatus used to determine s......
KNOWN: Operating conditions of apparatus used to determine surface boiling characteristics.
FIND: (a) Nucleate boiling coefficient for special coating, (b) Surface temperature as a function of heat flux; apparatus temperatures for a prescribed heat flux; applicability of nucleate boiling correlation for a specified heat flux.
ASSUMPTIONS: (1) One-dimensional, steady-state conduction in the bar, (2) Water is saturated at 1 atm, (3) Applicability of Rohsenow correlation with n = 1.
PROPERTIES: Table A.6, saturated water (100°C): ρ_{\ell} = 957.9 kg/m³, \mathrm c_{\mathrm p,\ell} = 4217 J/kg⋅K, μ_{\ell} = 279 × 10^{-6} N⋅s/m², \mathrm{Pr}_{\ell} = 1.76, \mathrm h_{\mathrm{fg}} = 2.257 × 10^6 J/kg, σ = 0.0589 N/m, ρ_{\mathrm v} = 0.5955 kg/m³.
SCHEMATIC:
ANALYSIS: (a) The coefficient \mathrm C_{\mathrm{s,f}} associated with Eq. 10.5 may be determined if \mathrm q_{\mathrm s}^{\prime\prime}~\text{and}~\mathrm T_{\mathrm s} are known. Applying Fourier’s law between \mathrm x_1~\text{and}~\mathrm x_2,
q _{ s }^{\prime \prime}= q _{\text {cond}}^{\prime \prime}= k \frac{ T _2 ~ – ~ T _1}{ x _2 ~- ~x _1}=400 W / m \cdot K \times \frac{(158.6 ~- ~ 133.7)^{\circ} C }{0.015 m }=6.64 \times 10^5 W / m ^2
Since the temperature distribution in the bar is linear, \mathrm T_{\mathrm s} = \mathrm T_1 – (\mathrm{dt/dx})\mathrm x_1 = \mathrm T_1 – [(\mathrm T_2 – \mathrm T_1)/(\mathrm x_2 – \mathrm x_1)] \mathrm x_1. Hence,
T _{ s }=133.7^{\circ} C -\left[24.9^{\circ} C / 0.015 m \right] 0.01 m =117.1^{\circ} C
From Eq. 10.5, with n = 1,
C _{ s , f }=\frac{ c _{ p , \ell} \Delta T _{ e }}{ h _{ fg } \operatorname{Pr}_{\ell}}\left\lgroup\frac{\mu_{\ell} h _{ fg }}{ q _{ s }^{\prime \prime}}\right\rgroup^{1 / 3}\left[\frac{ g \left(\rho_{\ell}~-~\rho_{ v }\right)}{\sigma}\right]^{1 / 6}
C _{ s, f } = \frac{4217 J / kg \cdot K \left(17.1^{\circ} C \right)}{2.257~ \times~ 10^6 J / kg (1.76)}\left\lgroup \frac{279~ \times~ 10^{-6} kg / s \cdot m ~\times ~2.257 ~\times ~10^6 J / kg }{6.64~ \times~ 10^5 W / m ^2}\right\rgroup^{1 / 3}\left[\frac{9.8 m / s ^2~ \times~ 957.3 kg / m ^3}{0.0589 kg / s ^2}\right]^{1 / 6}
C _{ s, f } = 0.0131
(b) Using the appropriate IHT Correlations and Properties Toolpads, the following portion of the nucleate boiling regime was computed.
For \mathrm q_{\mathrm s}^{\prime\prime} = 10^6 W/m² = \mathrm q_{\text{cond}}^{\prime\prime}, \mathrm T_{\mathrm s} = 119.6°C and
T_1 = 144.6^{\circ}C~~~~~\text{and}~~~~~~T_2 = 182.1^{\circ}C
With the critical heat flux given by Eq. 10.7,
q _{\max}^{\prime \prime}=0.149 h _{ fg } \rho_{ v }\left[\frac{\sigma g \left(\rho_{\ell} ~ -~ \rho_{ v }\right)}{\rho_{ v }^2}\right]^{1 / 4}
q _{\max }^{\prime \prime} = 0.149\left(2.257 \times 10^6 J / kg \right) 0.5955 kg / m ^3\left[\frac{0.0589 kg / s ^2 ~\times~ 9.8 m / s ^2 ~\times~ 957.3 kg / m ^3}{\left(0.5955 kg / m ^3\right)^2}\right]^{1 / 4}
q _{\max}^{\prime \prime} = 1.25 \times 10^6 W/m^2
Since \mathrm q_{\mathrm s}^{\prime\prime} = 1.5 \times 10^6 W/m² > \mathrm q_{\max}^{\prime\prime}, the heat flux exceeds that associated with nucleate boiling and the foregoing results can not be used.
COMMENTS: For \mathrm q_{\mathrm s}^{\prime\prime} > \mathrm q_{\max}^{\prime\prime}, conditions correspond to film boiling, for which \mathrm T_{\mathrm s} may exceed acceptable operating conditions.
