Question 12.113: KNOWN: Shallow pan of water exposed to night desert air and ......
KNOWN: Shallow pan of water exposed to night desert air and sky conditions.
FIND: Whether water will freeze.
ASSUMPTIONS: (1) Steady-state conditions, (2) Bottom of pan is well insulated, (3) Water surface is diffuse-gray, (4) Sky provides blackbody irradiation, \mathrm G_{\text{sky}} = σ\mathrm T^4_{\text{sky}}.
PROPERTIES: Table A-11, Water (300 K): ε = 0.96.
SCHEMATIC:
ANALYSIS: To estimate the water surface temperature for these conditions, begin by performing an energy balance on the pan of water considering convection and radiation processes.
\dot{ E }_{\text{in}}^{\prime \prime}-\dot{ E }_{\text{out}}^{\prime \prime}=0
\alpha G _{\text {sky}}-\varepsilon E _{ b }-\overline{ h }\left( T _{ s }- T _{\infty}\right)=0
\varepsilon \sigma\left( T _{\text{sky}}^4- T _{ s }^4\right)-\overline{ h }\left( T _{ s }- T _{\infty}\right)=0.
Note that, from Eq. 12.64, \mathrm G_{\text{sky}} = \sigma\mathrm T^4_{\text{sky}} and from Assumption 3, α = ε. Substituting numerical values, with all temperatures in kelvin units, the energy balance is
0.96 \times 5.67 \times 10^{-8} \frac{ W }{ m ^2 \cdot K ^4}\left[(-40+273)^4- T _{ s }^4\right] K ^4-5 \frac{ W }{ m ^2 \cdot K }\left[ T _{ s }-(20+273)\right] K =0
5.443 \times 10^{-8}\left[233^4- T _{ s }^4\right]-5\left[ T _{ s }-293\right]=0.
Using a trial-and-error approach, find the water surface temperature,
T_s = 268.5 K.
Since \mathrm T_{\mathrm s} < 273 K, it follows that the water surface will freeze under the prescribed air and sky conditions.
COMMENTS: If the heat transfer coefficient were to increase as a consequence of wind, freezing might not occur. Verify that for the given \mathrm T_∞~\text{and}~\mathrm T_{\text{sky}}, that if \overline{\mathrm h} increases by more than 40%, freezing cannot occur.