Two-Phase Heat Transfer and Critical Flow — Flow Boiling, Choking, and Pressure Drop Analysis

Scope: comprehensive treatment of two-phase flow thermohydraulics. Covers mixture formulations, void fraction and drift flux, two-phase pressure drop, flow boiling and condensation, and critical mass flux phenomena.

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1. Two-Phase Flow Modeling Framework

1.1 Definitions

• Quality (x): mass fraction of vapor $x = \frac{\dot{m}_v}{\dot{m}_l + \dot{m}_v}.$
• Void fraction (α): volumetric vapor fraction $\alpha = \frac{V_v}{V_l + V_v}.$
• Slip ratio (S): ratio of phase velocities $S = \frac{v_v}{v_l}.$

Relation between $x$and $\alpha$: $\alpha = \frac{x/\rho_v}{x/\rho_v + (1-x)/\rho_l S}.$

1.2 Mixture Properties

• Density: $\rho_m = \alpha \rho_v + (1-\alpha) \rho_l.$
• Viscosity (approx.): $\mu_m = \alpha \mu_v + (1-\alpha) \mu_l.$
• Specific enthalpy: $h_m = (1-x)h_l + xh_v.$

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2. Two-Fluid and Homogeneous Mixture Models

2.1 Two-Fluid (Separated) Model

Conservation equations written for each phase: $\partial(\alpha_k\rho_k)/\partial t + \nabla\cdot(\alpha_k\rho_k\mathbf{v}_k) = \Gamma_k,$ $\partial(\alpha_k\rho_k\mathbf{v}_k)/\partial t + \nabla\cdot(\alpha_k\rho_k\mathbf{v}_k\mathbf{v}_k) = -\alpha_k\nabla p + \nabla\cdot(\alpha_k\boldsymbol{\tau}_k) + \alpha_k\rho_k\mathbf{g} + \mathbf{M}_k.$ Interfacial momentum term $\mathbf{M}_k$provides coupling between phases.

2.2 Homogeneous Equilibrium Model (HEM)

Assumes: $v_v = v_l = v_m,\; T_v = T_l = T_{sat}$. Single mixture equation set: $\partial(\rho_m)/\partial t + \nabla\cdot(\rho_m\mathbf{v}_m) = 0.$ $\partial(\rho_m\mathbf{v}_m)/\partial t + \nabla\cdot(\rho_m\mathbf{v}_m\mathbf{v}_m) = -\nabla p + \nabla\cdot(\mu_m\nabla\mathbf{v}_m) + \rho_m\mathbf{g}.$

2.3 Homogeneous Nonequilibrium Model (HNE)

Includes relaxation between phase velocities and pressures — used near critical flow and flashing.

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3. Drift Flux Model

Proposed by Zuber & Findlay (1965) to correlate void fraction and phase velocities: $j = \alpha v_v + (1-\alpha)v_l = C_0 j_m + V_{gj},$ where:

• $C_0$: distribution parameter (1.0–1.3)
• $V_{gj}$: drift velocity (slip between phases)
• $j_m = G/\rho_m$: mixture velocity.

Void fraction: $\alpha = \frac{C_0 j_m + V_{gj}}{v_v + V_{gj}}.$

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4. Two-Phase Pressure Drop

Total pressure gradient: $\frac{dp}{dz} = \left(\frac{dp}{dz}\right)_{fric} + \left(\frac{dp}{dz}\right)_{grav} + \left(\frac{dp}{dz}\right)_{acc}.$

4.1 Gravitational Component

$\left(\frac{dp}{dz}\right)_{grav} = \rho_m g.$

4.2 Accelerational Component

$\left(\frac{dp}{dz}\right)_{acc} = G^2 \frac{d}{dz}\left(\frac{x^2}{\rho_v} + \frac{(1-x)^2}{\rho_l}\right).$

4.3 Frictional Component

Friction factor defined via two-phase multiplier $\Phi^2$: $\left(\frac{dp}{dz}\right)_{tp} = \Phi^2 \left(\frac{dp}{dz}\right)_{lo},$ where $(dp/dz)_{lo}$is the liquid-only pressure drop.

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5. Lockhart–Martinelli Correlation (1949)

Defines Martinelli parameter: $X = \left[\frac{(dp/dz)_{lo}}{(dp/dz)_{vo}}\right]^{1/2} = \left(\frac{\mu_l/\mu_v}{(\rho_v/\rho_l)^2}\frac{x^2}{(1-x)^2}\right)^{1/2}.$

Two-phase multipliers: $\Phi_l^2 = 1 + C/X + 1/X^2, \quad \Phi_v^2 = 1 + CX + X^2.$

Empirical constant C:

Flow type	C
Laminar–Laminar	5
Laminar–Turbulent	10
Turbulent–Turbulent	20

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6. Flow Boiling Heat Transfer

Heat transfer in two-phase forced convection arises from both convection and phase change: $q'' = h_{tp} (T_w - T_{sat}).$

6.1 Chen Correlation

Combines nucleate and convective contributions: $h_{tp} = S h_{pool} + F h_{conv}.$

Where:

• $S$: suppression factor (accounts for vapor turbulence)
• $F$: enhancement factor (two-phase convection)

6.2 Kandlikar Correlation (1983)

$h_{tp} = C_1 Re_l^{C_2} Bo^{C_3} (\frac{\rho_l}{\rho_v})^{C_4} k_l/D_h,$ where $Bo = q''/(Gh_{fg})$is the boiling number.

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7. Two-Phase Condensation Heat Transfer

Condensation in tubes may occur in annular or stratified flow.

7.1 Cavallini–Zecchin Correlation

$h_{tp} = h_{lo} (1 + 3.8 Bo^{0.76}),$ with $h_{lo}$as liquid-only Nusselt convection coefficient.

7.2 Film Condensation with Shear

$Nu_{tp} = 0.943 (Re_l Pr_l)^{1/3} (\rho_l/\rho_v)^{1/4}.$

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8. Critical Flow and Choking

Occurs when mass flux $G$reaches a maximum with respect to downstream pressure.

8.1 Isentropic Criterion

$\left(\frac{\partial G}{\partial p_2}\right)_{p_1} = 0.$

For compressible single-phase: $\dot{m} = \rho A c, \quad c = \sqrt{\gamma RT}.$

8.2 Homogeneous Equilibrium Model (HEM) Critical Flow

Assumes thermal and mechanical equilibrium: $G_{crit} = \sqrt{\frac{2(p_u - p_d)}{v_d^2 - v_u^2}}.$

For isentropic expansion: $G_{crit} \approx \left[ \frac{2\rho_l (p_u - p_v)}{1 + (\rho_l/\rho_v)(h_{fg}/c_l^2)} \right]^{1/2}.$

8.3 Nonequilibrium Flashing (HNE)

Includes metastable superheated liquid and relaxation: $G_{crit} = G_{HEM} (1 + K_{relax})^{-1/2}.$

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9. Empirical Critical Flow Correlations

Model	Formulation	Application
Moody (1965)	$G = C \rho_l (\Delta p/\rho_l)^{1/2}$	Short nozzles, water
Henry–Fauske (1971)	Semi-empirical, includes slip and non-equilibrium	Reactor safety
Leung–Moody	$G = K (p_{in}-p_{out})^{0.5}$	Flashing liquids

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10. Two-Phase Pressure Drop and Heat Transfer Maps

Lockhart–Martinelli coordinates used to correlate boiling and condensation transitions.

10.1 Flow Pattern Maps

Plot superficial velocities $j_g, j_l$:

• Bubbly → Slug → Annular → Mist

Transition lines defined by void fraction and Weber number criteria: $We = \frac{\rho U^2D}{\sigma}.$

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11. Entropy Generation and Efficiency

Entropy generation in two-phase flow: $\sigma_s = \sum_k \left( \frac{\boldsymbol{\tau}_k:\nabla v_k}{T_k} + \frac{k_k(\nabla T_k)^2}{T_k^2} + \frac{J_m\cdot\nabla\mu_k}{T_k} \right).$

Flow boiling irreversibility dominated by phase slip, viscous dissipation, and interfacial heat transfer across finite $\Delta T$.

Exergy destruction: $\dot{E}_D = T_0 \sigma_s V.$

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12. Typical Correlations Summary

Quantity	Correlation	Notes
Pressure drop	Lockhart–Martinelli, Friedel	Friction multiplier models
Heat transfer (boiling)	Chen, Kandlikar	Convective + nucleate terms
Condensation	Cavallini–Zecchin	Filmwise, annular
Critical flow	Henry–Fauske	Non-equilibrium flashing
Void fraction	Zuber–Findlay	Drift-flux model

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13. Cross-Links

• 07_Boiling_and_Condensation.md — detailed nucleate and film heat transfer mechanisms.
• 06_Multiphase_Flows_and_Bubble_Dynamics.md — phase transport and Rayleigh–Plesset formulation.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy production and coupling.
• Heat_Transfer/TwoPhase_Design_Methods.md — practical design and numerical methods for two-phase systems.