Boiling Condensation

Phenomenon

Boiling and Condensation — Phase Change Heat Transfer and Interfacial Thermodynamics

Scope: comprehensive treatment of liquid–vapor phase-change heat transfer, covering thermodynamic fundamentals, nucleate and film boiling, condensation, interfacial instabilities, critical heat flux, and entropy/exergy aspects.

1. Fundamentals of Phase Change

1.1 Saturation and Latent Heat

At equilibrium between liquid and vapor: psat(T)=pv=pl.p_{sat}(T) = p_v = p_l. The latent heat of vaporization hfg=hv−hlh_{fg} = h_v - h_l represents the energy absorbed during phase change at constant T,p.T,p.

1.2 Clapeyron Equation

Derived from Gibbs free energy equality across phases: dpsatdT=hfgT(vv−vl).\frac{dp_{sat}}{dT} = \frac{h_{fg}}{T(v_v - v_l)}. For vv≫vlv_v ≫ v_l: ln⁡psat=−hfgR1T+C.\ln p_{sat} = -\frac{h_{fg}}{R} \frac{1}{T} + C.

2. Boiling — Mechanisms and Regimes

Boiling is vapor formation within or at the surface of a liquid due to superheating above the saturation temperature.

2.1 Classification

TypeDriving conditionTypical location
Pool boilingFree convection, stationary liquidSurface-heated systems
Flow boilingForced convection, moving liquidTubes, channels, reactors

3. Microscopic Mechanism of Nucleate Boiling

3.1 Bubble Nucleation

A vapor embryo must overcome surface tension and pressure forces: Δp=pv−pl=2σrc.Δp = p_v - p_l = \frac{2σ}{r_c}. Critical superheat: ΔTc=Tsat2σhfgρvrc.ΔT_c = T_{sat}\frac{2σ}{h_{fg}ρ_v r_c}.

The activation barrier is the Gibbs free energy of formation: ΔG∗=16πσ33(Δp)2.ΔG^* = \frac{16πσ^3}{3(Δp)^2}.

3.2 Bubble Growth

Assuming heat transfer–controlled growth: R(t)≈2S√αlt,S=(Tw−Tsat)ρlhfgρvhfg.R(t) ≈ 2S√{α_l t}, \quad S = \frac{(T_w - T_{sat})ρ_l h_{fg}}{ρ_v h_{fg}}.

3.3 Departure and Detachment

Buoyancy and surface tension balance: Fb≈Fσ⇒Rd≈3σ2g(ρl−ρv).F_b ≈ F_σ \Rightarrow R_d ≈ \sqrt{\frac{3σ}{2g(ρ_l - ρ_v)}}.

Bubble departure frequency: f≈12πRdg(ρl−ρv)ρl.f ≈ \frac{1}{2πR_d} \sqrt{\frac{g(ρ_l - ρ_v)}{ρ_l}}.

4. Boiling Curve and Heat Flux Regimes

4.1 Regimes of Pool Boiling

RegimeMechanismTypical behavior
Natural convectionSingle-phase, no vaporq″ ∝ (ΔT)^n, n≈3
Nucleate boilingBubble growth, latent heatHigh heat transfer, stable
Transition boilingInstability, partial filmUnstable q″
Film boilingVapor layer insulationLow heat flux, high ΔT

4.2 Boiling Curve

A log–log plot of q″q″ vs. ΔT=Tw−TsatΔT = T_w - T_{sat} shows critical points:

  • Onset of Nucleate Boiling (ONB)
  • Critical Heat Flux (CHF)
  • Leidenfrost point

5. Nucleate Boiling Heat Transfer Correlations

5.1 Rohsenow Correlation (1952)

q″=μlhfg[g(ρl−ρv)σ]1/2[cpl(ΔT)3CsfhfgPrln],q″ = μ_l h_{fg} \left[ \frac{g(ρ_l - ρ_v)}{σ} \right]^{1/2} \left[ \frac{c_{pl}(ΔT)^3}{C_{sf} h_{fg} Pr_l^{n}} \right], where CsfC_{sf} depends on surface–fluid combination.

5.2 For Water on Metal Surfaces

Typical parameters: n=1,  Csf≈0.013–0.015.n = 1,\; C_{sf} ≈ 0.013–0.015.

6. Critical Heat Flux (CHF)

At the CHF, vapor blankets the surface, sharply reducing heat transfer — burnout.

6.1 Zuber’s Hydrodynamic Theory

Based on Kelvin–Helmholtz instability at the liquid–vapor interface: q″CHF=0.131hfgρv1/2[σg(ρl−ρv)]1/4.q″_{CHF} = 0.131 h_{fg} ρ_v^{1/2} [σg(ρ_l - ρ_v)]^{1/4}.

Occurs at wall superheat ~30–40 K for water at 1 atm.

6.2 Kutateladze Correlation (dimensionless form)

q″CHFρvhfg(gσ(ρl−ρv)/ρv2)1/4=0.16.\frac{q″_{CHF}}{ρ_v h_{fg} (gσ(ρ_l - ρ_v)/ρ_v^2)^{1/4}} = 0.16.

7. Film Boiling and the Leidenfrost Effect

At high surface temperatures, a stable vapor layer separates the surface from liquid.

7.1 Film Boiling Heat Flux

Approximate by conduction across vapor film: q″=kv(Tw−Tsat)δv.q″ = \frac{k_v (T_w - T_{sat})}{δ_v}.

Film thickness from force balance: δv∼(μv2hfgρvgkv(Tw−Tsat))1/3.δ_v ∼ \left( \frac{μ_v^2 h_{fg}}{ρ_v g k_v (T_w - T_{sat})} \right)^{1/3}.

7.2 Leidenfrost Temperature

Point of minimum heat flux: TLeid−Tsat≈150–200 K (for water).T_{Leid} - T_{sat} ≈ 150–200 \text{ K (for water)}.

8. Flow Boiling — Forced Convection with Phase Change

8.1 Two-Phase Flow Regimes

  • Bubbly/Slug: Discrete bubbles in liquid core
  • Annular: Continuous vapor core, liquid film on walls
  • Mist: Dispersed droplets in vapor stream

8.2 Flow Boiling Correlations

Chen (1966) correlation combines nucleate and convective components: q″=Sq″pool+Fq″conv,q″ = S q″_{pool} + F q″_{conv}, where SS and FF are suppression and enhancement factors depending on vapor quality and mass flux.

9. Condensation — Mechanisms and Regimes

Condensation is vapor-to-liquid phase change at sub-saturation wall temperatures.

9.1 Filmwise vs. Dropwise

TypeDescriptionHeat transfer coefficient
FilmwiseContinuous condensate filmLow, due to resistance
DropwiseDiscrete droplets, partial surface exposure5–10× higher

10. Nusselt Theory of Laminar Film Condensation (1916)

For condensation on a vertical plate: ddy(ρluhfg)=gρl(ρl−ρv)y.\frac{d}{dy}(ρ_l u h_{fg}) = gρ_l(ρ_l - ρ_v)y. Solution yields local film thickness: δ(y)=[4μlyq″ρlg(ρl−ρv)hfg]1/4.δ(y) = \left[ \frac{4μ_l y q″}{ρ_l g (ρ_l - ρ_v) h_{fg}} \right]^{1/4}.

Average heat transfer coefficient: hm=0.943[kl3ρl2ghfgμlL(Tsat−Tw)]1/4.h_m = 0.943 \left[ \frac{k_l^3 ρ_l^2 g h_{fg}}{μ_l L (T_{sat} - T_w)} \right]^{1/4}.

10.1 Effect of Vapor Shear

In forced convection condensation: Nux=0.943(RexPrl)1/3(ρl/ρv)1/4.Nu_x = 0.943 (Re_x Pr_l)^{1/3} (ρ_l/ρ_v)^{1/4}.

11. Dropwise Condensation

Droplet nucleation and growth controlled by surface wettability and condensation rate.

Heat transfer dominated by exposed area fraction fAf_A: hdrop≈fAhfilm1−fA.h_{drop} ≈ \frac{f_A h_{film}}{1 - f_A}.

Hydrophobic coatings promote dropwise condensation by reducing contact angle hysteresis.

12. Interfacial Stability and Wave Formation

Kelvin–Helmholtz instability at the liquid–vapor interface: ω2=gkρl−ρvρl+ρv+σk3ρl+ρv−(ρlρv)(ρl+ρv)2(ΔU)2k2.ω^2 = gk\frac{ρ_l - ρ_v}{ρ_l + ρ_v} + \frac{σk^3}{ρ_l + ρ_v} - \frac{(ρ_lρ_v)}{(ρ_l + ρ_v)^2} (ΔU)^2 k^2.

Instability occurs when shear ΔUΔU exceeds stabilizing surface tension and gravity effects.

13. Entropy Generation and Exergy in Phase Change

Local entropy production in boiling/condensation: σs=q″(1/Twall−1/Tsat)A+k(∇T)2T2+μ(∇v)2T.σ_s = \frac{q″(1/T_{wall} - 1/T_{sat})}{A} + \frac{k(∇T)^2}{T^2} + \frac{μ(∇v)^2}{T}.

Exergy destruction rate: E˙D=T0σs.\dot{E}_D = T_0 σ_s.

Efficiency of phase-change heat transfer is limited by irreversibility from finite temperature differences and non-equilibrium phase interfaces.

14. Summary Correlations

PhenomenonCorrelationKey Dependencies
Rohsenow (pool boiling)q″∝(ΔT)3q″ ∝ (ΔT)^3Surface, Prandtl number
Zuber CHFq″CHF=0.131hfgρv1/2[σg(ρl−ρv)]1/4q″_{CHF} = 0.131h_{fg}ρ_v^{1/2}[σg(ρ_l - ρ_v)]^{1/4}Instability-limited
Nusselt condensationhm=0.943[kl3ρl2ghfg/(μlLΔT)]1/4h_m = 0.943[k_l^3ρ_l^2gh_{fg}/(μ_lLΔT)]^{1/4}Film conduction
Dropwise condensationh≈(fA/(1−fA))hfilmh ≈ (f_A/(1-f_A))h_{film}Wettability
  • 06_Multiphase_Flows_and_Bubble_Dynamics.md — nucleation, bubble growth, and Rayleigh–Plesset theory.
  • Thermodynamics/09_Phase_Transitions_and_Critical_Phenomena.md — Clapeyron relation and criticality.
  • Heat_Transfer/Condensation_Correlation_Models.md — detailed numerical formulations.
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy production and irreversibility.