Question 10.19: KNOWN: Kutateladze’s dimensional analysis and the bubble dia......

ANALYSIS: (a) Kutateladze postulated that the critical flux was dependent upon four parameters,

q _{\max }^{\prime \prime}= q _{\max }^{\prime \prime}\left( h _{ fg }, \rho_{ v }, \sigma, D _{ b }\right)

where \mathrm D_{\mathrm b} is the bubble diameter parameter having the form

D _{ b }=\left[\sigma / g \left(\rho_{\ell}-\rho_{ v }\right)\right]^{1 / 2}.                                                                                    (1)

The form of the critical heat flux expression was presumed to be

q _{\max }^{\prime \prime}= C  h _{ fg }  \rho_{ v }^{1 / 2}  D _{ b }^{-1 / 2}  \sigma^{1 / 2}                                                                                    (2)

where C is a constant. It is not possible to derive this equation from a dimensional (Pi) analysis. We can only determine that the equation is dimensionally consistent. Using SI units, check Eq. (1) for \mathrm D_{\mathrm b},

D _{ b } \Rightarrow \left[\left( Nm ^{-1}\right)\left( m ^{-1} s ^2\right)\left( kg ^{-1} m ^3\right)\right]^{1 / 2} \Rightarrow\left[ N \left\lgroup \frac{ s ^2}{ kg \cdot m ^2}\right\rgroup m ^3\right]^{1 / 2} \Rightarrow[ m ]

and in Eq. (2) for \mathrm q_{\max}^{\prime\prime},

q _{\max }^{\prime \prime} \Rightarrow \left[\left( Jkg ^{-1}\right)\left( kg ^{1 / 2} m ^{-3 / 2}\right)\left( m ^{-1 / 2}\right)\left( N ^{1 / 2} m ^{-1 / 2}\right)\right] \Rightarrow\left[\frac{ J }{ s } \cdot\left\lgroup \frac{ N \cdot s ^2}{ kg \cdot m }\right\rgroup^{1 / 2} m ^{-2}\right] \Rightarrow\left[\frac{ W }{ m ^2}\right].

Hence, the equations are dimensionally consistent.

(b) The Bond number, Bo, is defined as the ratio of the characteristic length L (width or diameter) of the heater surface to the bubble diameter parameter, \mathrm D_{\mathrm b}. That is, Bo ≡ L/\mathrm D_{\mathrm B}. The number squared is also indicative of the ratio of the buoyant to capillary forces. For water at 1 atm (see Example 10.1 for properties listing), Eq. (1) yields

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}=0.0025  m =2.5  mm.

Eq. 10.7 for the critical heat flux is appropriate for an “infinite” heater (Bo ≥ 3). To meet this requirement, the heater dimension must be

L \geq Bo \cdot D _{ b }=3 \times 2.5  mm =7.5  mm.

COMMENTS: As the heater size decreases (Bo decreasing), the boiling curve no longer exhibits the characteristic \mathrm q_{\max}^{\prime\prime}~\text{and}~\mathrm q_{\min}^{\prime\prime} features. The very small heater, such as a wire, is enveloped with vapor at small \Delta\mathrm T_{\mathrm e} and film boiling occurs.