Question

For laminar film condensation on vertical surface, the film thickness:

• A. cumulatively decreases from top to bottom
• B. cumulatively increases from top to bottom
• C. the surface conductance increase from top to bottom
• D. remains constant from top to bottom

Hint:

Remember the core trends for Nusselt film condensation on vertical surfaces: - Film thickness: \( \delta(x) \propto x^{1/4} \) (Increases downstream from top to bottom). - Heat transfer coefficient: \( h(x) \propto x^{-1/4} \) (Decreases downstream from top to bottom). The film grows thicker because gravity drags the existing fluid down while fresh vapor continuously condenses and appends onto it.

Answer 1

The Correct Option is B

Concept: Laminar film condensation along an extended vertical solid surface is comprehensively analyzed utilizing the classical Nusselt model of condensation. When a cold vertical plate is exposed to a pure saturated vapor, condensation initiates at the topmost edge of the plate (\( x = 0 \)). As the condensed liquid moves downward under the driving influence of gravity, the mass flow rate of the liquid film continuously increases because more vapor condenses onto the existing liquid layer along the vertical descent path.

Step 1: Understanding Nusselt's Analytical Solution for Film Thickness.
According to Nusselt's boundary layer analysis for a vertical orientation plate, the local physical thick profile thickness of the continuous condensate film (denoted by \( \delta(x) \)) at a downward vertical distance \( x \) measured directly from the top entry edge is governed by the following mathematical formula: \[ \delta(x) = \left[ \frac{4 \cdot \mu \cdot k \cdot (T_{sat} - T_w) \cdot x}{g \cdot \rho_l \cdot (\rho_l - \rho_v) \cdot h_{fg}^*} \right]^{1/4} \] Where:
• \( x \) is the localized vertical distance tracking from the top absolute margin downwards.
• \( \mu \) is the dynamic viscosity of the liquid film layer.
• \( k \) is the thermal conductivity behavior of the liquid phase.
• \( T_{sat} - T_w \) represents the temperature driving potential.
• \( g \) represents the local gravitational acceleration constant.
• \( \rho_l \) and \( \rho_v \) represent the mass densities of the liquid and vapor phases respectively.
• \( h_{fg}^* \) is the modified latent heat of vaporization.

Step 2: Analysis of the Spatial Functional Dependence.
By grouping all the constant physical fluid properties and system operating parameters together into a single constant lumped coefficient \( C \), the relationship simplifying spatial variation reduces to: \[ \delta(x) = C \cdot x^{1/4} \] This expression directly reveals that the thickness parameter \( \delta \) scales as a non-linear power-law function proportional to the fourth root of the downward vertical coordinates: \[ \delta(x) \propto x^{1/4} \] Evaluating this continuous behavior at key critical spatial points:
• At the extreme top edge where \( x = 0 \): The film layer thickness is zero, \( \delta(0) = 0 \).
• As the position coordinate increases moving downward (\( x \rightarrow \text{bottom} \)): The value of \( x^{1/4} \) grows monotonically. Therefore, the liquid condensate layer accumulates continuously as it drains down the vertical surface under the influence of gravity. This accumulation results in a film layer thickness that cumulatively and steadily increases from the top edge to the bottom boundary.

Step 3: Evaluating the Impact on Surface Heat Transfer Conductance.
The local convective heat transfer coefficient or surface conductance (denoted by \( h_x \)) is inversely proportional to the liquid film boundary thickness, as heat must conduct across this liquid layer: \[ h_x = \frac{k}{\delta(x)} \propto x^{-1/4} \] Because the localized film layer grows thicker downstream, the structural thermal resistance increases. Consequently, the local surface conductance \( h_x \) exhibits a cumulative decrease from top to bottom, making statement (3) incorrect. Thus, statement (2) is the uniquely correct choice.