Question 10.34: KNOWN: Surface temperature and emissivity of strip steel. FI......
KNOWN: Surface temperature and emissivity of strip steel.
FIND: Heat flux across vapor blanket.
ASSUMPTIONS: (1) Steady-state conditions, (2) Vapor/jet interface is at \mathrm T_{\text{sat}} for p = 1 atm, (3) Negligible effect of jet and strip motion.
PROPERTIES: Table A-6, Saturated water (100°C): ρ_{\ell} = 957.9 kg/m³, \mathrm h_{\mathrm{fg}} = 2257 kJ/kg; Saturated water vapor (\mathrm T_{\mathrm f} = 640 K): \rho_{\mathrm v} = 175.4 kg/m³, \mathrm c_{\mathrm{p,v}} = 42 kJ/kg·K, \mu_{\mathrm v} = 32 × 10^{-6} N·s/m², k = 0.155 W/m·K, \nu_{\mathrm v} = 0.182 × 10^{-6} m²/s.
SCHEMATIC:
ANALYSIS: The heat flux is
q_s^{\prime\prime} = \overline{h}\Delta T_e
where \Delta T_e = 907 K – 373 K = 534 K
and \overline{ h }^{4 / 3}=\overline{ h }_{\text {conv }}^{4 / 3}+\overline{ h }_{ rad } \overline{ h }^{ 1 / 3} ~~~\quad \text { or } \quad ~~~\overline{ h }=\overline{ h }_{\text {conv }}+(3 / 4) \overline{ h }_{ rad }.
With h _{ fg }^{\prime}= h _{ fg }+0.80 c _{ p , v }\left( T _{ s } – T _{ sat }\right)=2.02 \times 10^7 J / kg
Equation 10.9 yields
\overline{ Nu }_{ D }=0.62\left[\frac{9.8 m / s ^2(957.9 ~-~ 175.4) kg / m ^3\left(2.02 ~\times ~10^7 J / kg \right)(1 m )^3}{0.182~ \times~ 10^{-6} m ^2 / s (0.155 W / m \cdot K )(907 ~-~ 373) K }\right]^{1 / 4}=6243.
Hence,
\overline{ h }_{\text{conv}}=\overline{ Nu }_{ D } k _{ v } / D =6243 W / m ^2 \cdot K (0.155 W / m \cdot K / 1 m )=968 W / m ^2 \cdot K
\overline{ h }_{ rad }=\frac{\varepsilon \sigma\left( T _{ s }^4 ~ – ~ T _{ sat }^4\right)}{ T _{ s } ~ – ~ T _{ sat }}=\frac{0.35~ \times ~5.67 ~\times ~10^{-8} W / m ^2 \cdot K ^4\left(907^4 ~- ~373^4\right) K ^4}{(907 ~ – ~ 373) K }
\overline{ h }_{ rad }=24 W / m ^2 \cdot K
Hence, \overline{ h }=968 W / m ^2 \cdot K +(3 / 4)\left(24 W / m ^2 \cdot K \right)=986 W / m ^2 \cdot K
And q _{ s }^{\prime \prime}=986 W / m ^2 \cdot K (907 – 373) K =5.265 \times 10^5 W / m ^2.
COMMENTS: The foregoing analysis is a very rough approximation to a complex problem. A more rigorous treatment is provided by Zumbrunnen et al. In ASME Paper 87-WA/HT-5.