Film condensation on a vertical surface — The average heat-transfer coefficient for laminar film condensation varies with the condensate film temperature drop ΔT as:

Introduction:
Nusselt’s analysis for laminar film condensation on a vertical plate provides a closed-form scaling for the average heat-transfer coefficient. It reveals how physical properties and the temperature driving force influence condensate film thickness and heat transfer.

Given Data / Assumptions:

• Steady, laminar, gravity-driven condensate film.
• Negligible vapor shear, isothermal wall, constant properties.
• Temperature drop across the liquid film is ΔT = Tsat − Twall.

Concept / Approach:

The classical result gives h_avg ∝ [ (ρ_l (ρ_l − ρ_v) g k_l^3 h_fg) / (μ_l L ΔT) ]^(1/4). Because ΔT appears in the denominator inside the brackets, the overall dependence is h ∝ (ΔT)^(−1/4), i.e., inversely proportional to ΔT raised to the one-fourth power. Larger ΔT slightly thickens the film (via increased condensate flow), which reduces h modestly in this laminar regime.

Step-by-Step Solution:

Verification / Alternative check:

Comparisons with experimental data validate the weak inverse quarter-power dependence over the laminar range; deviations arise with vapor shear or turbulence in the film.

Why Other Options Are Wrong:

A/B/D predict much stronger inverse dependences than theory supports; E is incorrect since ΔT does influence h via the film flow rate.

Common Pitfalls:

Confusing condensation with convection where h often grows with ΔT; here, film dynamics impose the inverse relationship.

Final Answer:

(ΔT)^(1/4) (inversely proportional)