Question

The heat flux in the nucleate boiling regime is proportional to

• A. \((\Delta T)^2\)
• B. \((\Delta T)^4\)
• C. \((\Delta T)^3\)
• D. \(\sqrt{\Delta T}\)

Hint:

The exponent of \(\Delta T\) in nucleate boiling correlations is typically around 3, indicating a strong dependence of heat flux on the excess temperature. This high dependence is what makes nucleate boiling a very efficient heat transfer mechanism.

Answer 1

The Correct Option is C

Step 1: Understand the nucleate boiling regime.
Nucleate boiling is a type of boiling that occurs when the surface temperature is slightly above the saturation temperature of the liquid (i.e., small excess temperature, \(\Delta T = T_{surface} - T_{saturation}\)). It is characterized by the formation of bubbles at nucleation sites on the heated surface. These bubbles grow and detach, causing significant mixing of the fluid near the surface and a very high rate of heat transfer. The heat flux (\(q''\)) in the nucleate boiling regime is strongly dependent on the excess temperature (\(\Delta T\)). Step 2: Recall the empirical correlations for heat flux in nucleate boiling.
While a precise theoretical derivation for the relationship between heat flux and excess temperature in nucleate boiling is complex due to the many factors involved (surface roughness, nucleation site density, fluid properties, etc.), empirical correlations have been developed based on experimental data. One of the most well-known and widely used correlations for nucleate pool boiling heat flux is given by Rohsenow's correlation: $q'' = \mu_l h_{fg} \left( \frac{g( \rho_l - \rho_v )}{\sigma} \right)^{\frac{1}{2}} \left( \frac{c_{p,l} \Delta T}{C_{sf} h_{fg} Pr_l^n} \right)^3$ Where:
\(q''\) is the heat flux.
\(\mu_l\) is the liquid viscosity.
\(h_{fg}\) is the latent heat of vaporization.
\(g\) is the acceleration due to gravity.
\(\rho_l\) is the liquid density.
\(\rho_v\) is the vapor density.
\(\sigma\) is the surface tension.
\(c_{p,l}\) is the specific heat of the liquid.
\(\Delta T = T_{surface} - T_{saturation}\) is the excess temperature.
\(C_{sf}\) is a surface-fluid combination constant.
\(Pr_l\) is the Prandtl number of the liquid.
\(n\) is an exponent that depends on the surface-fluid combination (typically around 1.0 for water and 1.7 for other fluids).
Step 3: Identify the proportionality between heat flux and \(\Delta T\) from the correlation.
From Rohsenow's correlation, we can observe that the heat flux \(q''\) is proportional to the cube of the excess temperature (\(\Delta T\)), raised to the power of 3. The other terms in the correlation (\(\mu_l, h_{fg}, g, \rho_l, \rho_v, \sigma, c_{p,l}, C_{sf}, Pr_l, n\)) are properties of the fluid and the surface and are considered constant for a given boiling scenario. Therefore, the heat flux in the nucleate boiling regime is approximately proportional to \((\Delta T)^3\). $$q'' \propto (\Delta T)^3$$ Step 4: Match the proportionality with the given options.
The derived proportionality \(q'' \propto (\Delta T)^3\) matches option (3).