Question 10.20: KNOWN: Lienhard-Dhir critical heat flux correlation for smal......
KNOWN: Lienhard-Dhir critical heat flux correlation for small horizontal cylinders.
FIND: Critical heat flux for 1 mm and 3 mm diameter horizontal cylinders in water at 1 atm.
ASSUMPTIONS: Nucleate pool boiling.
PROPERTIES: Table A-6, Water (1 atm): ρ_{\ell} = 957.9 kg/m³, ρ_{\mathrm v} = 0.5955 kg/m³, σ = 58.9 × 10^{-3} N/m.
SCHEMATIC:
ANALYSIS: The Lienhard-Dhir correlation for small horizontal cylinders is
q _{\max }^{\prime \prime}= q _{\max,\mathrm Z}^{\prime \prime}\left[0.94( Bo )^{-1 / 4}\right] ~~~\quad~~~ 0.15 \leq Bo \leq 1.2 (1)
where \mathrm q_{\max,\mathrm Z}^{\prime\prime} is the critical heat flux predicted by the Zuber-Kutateladze correlation for the infinite heater (Eq. 10.6) and the Bond number is
Bo =\frac{ r }{ D _{ b }}= r /\left[\sigma / g \left(\rho_{\ell}-\rho_{ v }\right)\right]^{1 / 2}. (2)
Note the characteristic length is the cylinder radius. From Example 10.1, using Eq. 10.6,
q _{\max,\mathrm Z}^{\prime \prime} = 1.11 MW/m^2
and substituting property values for water at 1 atm into Eq. (2),
D _{ b }=\left[58.9 \times 10^{-3} \frac{ N }{ m } / 9.8 \frac{ m }{ s ^2}(957.9 – 0.5955) \frac{ kg }{ m ^3}\right]^{1 / 2}=2.51 mm.
Substituting appropriate values into Eqs. (1) and (2),
1 mm dia cylinder Bo = 0.5 mm/2.51 mm = 0.20
q _{\max }^{\prime \prime}=1.11 MW / m ^2\left[0.94(0.20)^{-1 / 4}\right]=1.56 MW / m ^2.
3 mm dia cylinder Bo = 1.5 mm/2.51 mm = 0.60
q _{\max }^{\prime \prime}=1.11 MW / m ^2\left[0.94(0.60)^{-1 / 4}\right]=1.19 MW / m ^2.
Note that for the 3 mm diameter cylinder, the critical heat flux is 1.19/1.11 = 1.07 times larger than the value for a very large horizontal cylinder.
COMMENTS: For practical purposes a horizontal cylinder of diameter greater than 3 mm can be considered as a very large one. The critical heat flux for a 1 mm diameter cylinder is 40% larger than that for the large cylinder.