Film and dropwise condensation, Nusselt theory

Film and dropwise condensation, Nusselt's laminar film theory for vertical plates and horizontal tubes, tube-bank and subcooling corrections, and design scaling.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Steam heaters, distillation-column condensers, surface condensers of power plants and refrigerant condensers all work by condensation. Nusselt's film theory tells you the condensing coefficient, how it depends on tube orientation, height and temperature difference, and why most process condensers use horizontal tubes.

Key ideas

When condensation happens. Vapour condenses on a surface whose temperature T_s is below the saturation temperature T_sat at the vapour pressure. The latent heat released must be conducted through the liquid to the wall.

Film condensation. When the liquid wets the surface (clean metals, most practical cases) the condensate forms a continuous film that drains under gravity. The film is the main resistance. On a vertical surface the film is zero at the top and thickens downward, δ ∝ x^(1/4), so the local coefficient falls as x^(−1/4).

Dropwise condensation. On a non-wetting surface (promoters such as fatty acids, polymer or noble-metal coatings) the condensate forms drops that grow, merge and roll off, leaving bare metal. Coefficients are 5–10 times higher than film condensation. However, promoters wash off or foul with time, so industrial condensers are designed conservatively for film condensation.

Nusselt's assumptions (vertical plate).

  1. Laminar film flow, driven by gravity, resisted by viscosity; inertia of the film neglected.
  2. Pure vapour at T_sat; no non-condensable gases.
  3. Negligible shear at the liquid–vapour interface (quiescent vapour).
  4. Constant wall temperature and constant liquid properties.
  5. Linear temperature profile across the film (heat crosses by conduction only).
  6. Subcooling of the condensate neglected; this is corrected by using a modified latent heat h′_fg = h_fg + 0.68·c_pl·(T_sat − T_s). Properties are taken at the film temperature (T_sat + T_s)/2, and h_fg at T_sat.

Results and scaling. The average coefficient for a vertical plate or tube of height L is h̄ = 0.943·[ρ_l(ρ_l − ρ_v)·g·h_fg·k_l³/(μ_l·L·ΔT)]^(1/4). So h̄ ∝ L^(−1/4) and ∝ ΔT^(−1/4): a larger temperature difference lowers h (more condensate, thicker film) even though the flux rises as ΔT^(3/4).

Horizontal tubes. The same theory gives the constant 0.725 with the tube diameter D in place of L. For a typical tube (L/D ≈ 40) the horizontal arrangement gives about twice the coefficient of the vertical one, which is why shell-side condensers usually have horizontal tubes. For a vertical bank of N tubes, condensate from upper tubes thickens the film on lower ones: h̄_N = h̄_1·N^(−1/4) (conservative).

Film Reynolds number. Re_δ = 4Γ/μ_l, with Γ the condensate flow per unit width (kg/s·m), or per unit perimeter for a vertical tube. Below about 30 the film is smooth laminar (Nusselt theory holds); 30–1800 it is wavy laminar (Nusselt underestimates h by up to about 20 %); above 1800 it is turbulent and h rises with Re_δ.

Non-condensable gases. Even a few percent of air accumulates at the interface and must diffuse away, cutting h drastically; condensers need vents.

Formulas

h̄_vert = 0.943·[ρ_l·(ρ_l − ρ_v)·g·h′_fg·k_l³/(μ_l·L·(T_sat − T_s))]^(1/4) Vertical plate or tube, laminar; ρ (kg/m³), g (m/s²), h′_fg (J/kg), k_l (W/m·K), μ_l (Pa·s), L height (m), temperatures (K or °C).

h̄_horiz = 0.725·[ρ_l·(ρ_l − ρ_v)·g·h′_fg·k_l³/(μ_l·D·(T_sat − T_s))]^(1/4) Single horizontal tube; D outside diameter (m).

h̄_N = h̄_1·N^(−1/4) Average for a vertical column of N horizontal tubes.

h′_fg = h_fg + 0.68·c_pl·(T_sat − T_s) Modified latent heat (J/kg); c_pl liquid specific heat (J/kg·K).

q = h̄·A·(T_sat − T_s), ṁ = q/h′_fg Heat rate (W), condensate rate (kg/s).

Re_δ = 4·ṁ/(μ_l·b) Film Reynolds number; b width of plate or πD for a vertical tube (m).

δ(x) = [4·μ_l·k_l·(T_sat − T_s)·x/(g·ρ_l·(ρ_l − ρ_v)·h′_fg)]^(1/4) Local film thickness (m) at distance x from the top.

Worked examples

Example 1 (standard). Saturated steam at 1 atm (100 °C) condenses on a vertical plate 0.3 m high and 1 m wide held at 90 °C. Liquid properties at 95 °C: ρ_l = 961.9 kg/m³, μ_l = 2.97 × 10⁻⁴ Pa·s, k_l = 0.677 W/m·K, c_pl = 4212 J/kg·K; ρ_v = 0.596 kg/m³, h_fg = 2257 kJ/kg. Find h̄, the heat rate and the condensate rate; check the regime.

  1. h′_fg = h_fg + 0.68·c_pl·ΔT = 2.257 × 10⁶ + 0.68 × 4212 × 10 = 2.2856 × 10⁶ J/kg.
  2. Bracket: 961.9 × 961.3 × 9.81 × 2.2856 × 10⁶ × 0.677³/(2.97 × 10⁻⁴ × 0.3 × 10) = 7.220 × 10¹⁵.
  3. h̄ = 0.943 × (7.220 × 10¹⁵)^(1/4) = 0.943 × 9218 = 8693 W/m²·K (≈ 8.69 kW/m²·K).
  4. q = h̄·A·ΔT = 8693 × 0.3 × 10 = 26 080 W.
  5. ṁ = q/h′_fg = 26 080/2.2856 × 10⁶ = 0.0114 kg/s.
  6. Re_δ = 4ṁ/(μ_l·b) = 4 × 0.0114/(2.97 × 10⁻⁴ × 1) = 154 → wavy laminar, so the Nusselt result is slightly conservative.

Answer: h̄ ≈ 8.7 kW/m²·K, q ≈ 26.1 kW, ṁ ≈ 0.0114 kg/s (41 kg/h).

Example 2 (GATE level). The same steam condenses on a 25 mm OD tube, 1 m long, with the wall at 90 °C. Compare the average coefficients with the tube horizontal and vertical, find the condensation rate for the horizontal tube, and the average h for a vertical column of 4 such horizontal tubes.

  1. Ratio from the two formulas: h̄_horiz/h̄_vert = (0.725/0.943)·(L/D)^(1/4) = 0.7688 × 40^(1/4) = 0.7688 × 2.515 = 1.93.
  2. Horizontal: h̄ = 0.725 × [same numerator/(μ_l·D·ΔT)]^(1/4) = 12 440 W/m²·K. Vertical (L = 1 m): 6433 W/m²·K.
  3. q = h̄·π·D·L·ΔT = 12 440 × π × 0.025 × 1 × 10 = 9770 W.
  4. ṁ = 9770/2.2856 × 10⁶ = 4.27 × 10⁻³ kg/s = 15.4 kg/h.
  5. Four tubes in a vertical column: h̄_4 = 12 440 × 4^(−1/4) = 12 440 × 0.707 = 8795 W/m²·K.

Answer: horizontal ≈ 12.4 kW/m²·K vs vertical ≈ 6.4 kW/m²·K (ratio 1.93); ≈ 15.4 kg/h condensed; 4-tube column average ≈ 8.8 kW/m²·K.

Common mistakes

  • Using ΔT in the numerator: h falls as ΔT^(−1/4).
  • Using the plate height L in the horizontal-tube formula instead of D, or the wrong constant (0.943 vs 0.725).
  • Evaluating h_fg at the film temperature instead of T_sat, or forgetting the subcooling correction.
  • Believing dropwise condensation is the design basis for industrial condensers.
  • Ignoring non-condensable gases, which can reduce the coefficient severalfold.
  • Saying the film has uniform thickness; it grows as x^(1/4).

For GATE CH

Expect ratio questions (how h̄ changes with height, ΔT, diameter, or horizontal vs vertical orientation), direct evaluation of Nusselt's formulas with given properties, condensate rate from a heat balance, the N^(−1/4) tube-bank correction, and conceptual questions on dropwise vs film condensation and non-condensables. Practise writing h ∝ L^(−1/4)·ΔT^(−1/4) and q ∝ L^(3/4)·ΔT^(3/4).

Quick check

  1. In laminar film condensation on a vertical plate, how does h̄ change if the height is increased 16 times?
  2. Which gives the higher h for a typical long tube: horizontal or vertical?
  3. Why do condensers need vents?
  4. What is the average h for a vertical column of 16 horizontal tubes relative to a single tube?
  5. What does the 0.68·c_pl·ΔT term account for?

Answers: 1. It halves. 2. Horizontal. 3. To remove non-condensable gases that blanket the surface. 4. 16^(−1/4) = 0.5 times. 5. Subcooling of the condensate film.

Interview questions

All Heat Transfer interview questions

Try answering each one aloud before you open it.

  1. 1.What is film condensation and how does it differ from dropwise condensation?Concept

    Film condensation occurs when a continuous film of condensate forms on a surface, which acts as a thermal resistance to heat transfer. Dropwise condensation, on the other hand, occurs when droplets form on the surface, allowing for more efficient heat transfer as the droplets can easily roll off, exposing more surface area. Dropwise condensation is generally more efficient than film condensation.

  2. 2.Explain Nusselt's theory of film condensation.Concept

    Nusselt modelled a laminar condensate film draining under gravity down a vertical wall held below T_sat, with pure quiescent vapour, no interfacial shear, constant properties and a linear temperature profile across the film. A force balance gives the velocity profile, a mass balance gives the film flow, and an energy balance (latent heat conducted through the film) shows the film thickness grows as x^(1/4) from the top. The result is h̄ = 0.943[ρ_l(ρ_l − ρ_v)g·h_fg·k_l³/(μ_l·L·ΔT)]^(1/4) for a vertical surface and 0.725 with D in place of L for a horizontal tube.

  3. 3.Dropwise condensation gives much higher coefficients. Why are industrial condensers still designed for film condensation?Application

    Dropwise condensation, where drops form and roll off leaving bare metal, can give coefficients 5–10 times higher than film condensation. But it needs a non-wetting surface, and promoters (oils, fatty acids, thin polymer coatings) wash off, oxidise or foul, while durable noble-metal coatings are expensive. Clean metal surfaces revert to film condensation within weeks, so designers use Nusselt film coefficients for a safe, conservative design and treat any dropwise contribution as a bonus. In any case the condensing side is rarely the controlling resistance once cooling-water and fouling resistances are included.

  4. 4.What factors can influence the transition from dropwise to film condensation?Application

    The transition from dropwise to film condensation can be influenced by surface properties such as wettability, surface roughness, and the presence of impurities or surfactants. A hydrophilic surface tends to promote film condensation, while a hydrophobic surface supports dropwise condensation. Additionally, the presence of contaminants can alter the surface energy, leading to a change in condensation mode.

  5. 5.What role does surface tension (wettability) play in condensation?Application

    Surface tension does not appear in Nusselt's film theory, which balances only gravity, viscosity and conduction. Its main role is through wettability: the balance of liquid–vapour, solid–liquid and solid–vapour surface energies sets the contact angle. A low contact angle (wetting, as on clean metal) spreads the condensate into a continuous film; a high contact angle (non-wetting or hydrophobic coatings) makes it bead up into drops, giving dropwise condensation. On finned or fluted tubes surface tension is also used to pull condensate off the fin tips, thinning the film there.

  6. 6.What happens if the surface temperature is below the saturation temperature during condensation?Application

    If the surface temperature is below the saturation temperature, condensation will occur as the vapor comes into contact with the cooler surface. The vapor will release latent heat and change phase into a liquid. The rate of condensation will depend on the temperature difference between the vapor and the surface, as well as the heat transfer coefficient.

  7. 7.Using Nusselt's theory, estimate the average condensing coefficient on a vertical plate 1 m high with h_fg = 2.26 × 10⁶ J/kg, ρ_l = 958 kg/m³, μ_l = 0.001 Pa·s, k_l = 0.6 W/m·K and T_sat − T_s = 10 K (neglect vapour density).Numerical

    Nusselt's result is h̄ = 0.943[ρ_l²·g·h_fg·k_l³/(μ_l·L·ΔT)]^(1/4). The bracket is 958² × 9.81 × 2.26 × 10⁶ × 0.216/(0.001 × 1 × 10) ≈ 4.39 × 10¹⁴, whose fourth root is about 4579. So h̄ ≈ 0.943 × 4579 ≈ 4.32 × 10³ W/m²·K. (A Grashof-number correlation is not used — that is for natural convection without phase change.)

  8. 8.What is the role of surface coatings in promoting dropwise condensation?Application

    Surface coatings can enhance dropwise condensation by altering the surface energy and wettability of the material. Hydrophobic coatings reduce the surface energy, making it less likely for a continuous film to form. This encourages the formation of droplets, which can easily roll off, maintaining a high heat transfer rate. Coatings must be durable and resistant to wear to maintain their effectiveness over time.

  9. 9.Explain how the heat transfer coefficient changes with the thickness of the condensate film.Concept

    The heat transfer coefficient is inversely related to the thickness of the condensate film. As the film thickness increases, the thermal resistance to heat transfer also increases, leading to a decrease in the heat transfer coefficient. This is because a thicker film acts as an insulating layer, reducing the rate of heat transfer from the vapor to the surface.

  10. 10.A vertical plate is experiencing laminar film condensation. If the plate height is doubled, how do the average coefficient and the heat transfer rate change according to Nusselt's theory?Numerical

    Nusselt's theory gives h̄ ∝ L^(−1/4), because the film keeps thickening down the plate. Doubling L multiplies h̄ by 2^(−1/4) ≈ 0.84. The heat transfer rate is h̄ × area × ΔT, and the area doubles, so q ∝ L^(3/4) and rises by 2^(3/4) ≈ 1.68 (assuming the film stays laminar).

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousHeat transfer with phase change: boilingNext Radiation laws: Stefan-Boltzmann, Kirchhoff and Wien