A SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations Herv´ e Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40 herve.lemonnier@cea.fr , herve.lemonnier.sci.free.fr/TPF/TPF.htm ECP, 2011-2012
DERIVATION OF CONTINUUM MECHANICS BALANCE EQUATIONS 1. First principles (4) • Leibniz rule and Gauss theorem. • On material and arbitrary control volumes. 2. Local instantaneous balance equations (single-phase). The closure issue (I) • Fixed volume with an interface (discontinuity surface). 3. Local instantaneous balance for each phase and the interface (jump condi- tions). • Space averaging: 1D balance equations. • Time averaging: 3D local balance equations (Reynolds style). • Composite averaging: two-fluid model. 4. The closure issue (II) Two-phase flow balance equations 1/39
� � � � � � � � � MATHEMATICAL TOOLS � � � • Displacement velocity of a surface, S : � ∂ M � � � � � � � � v S � ∂t u,v � � � • Depends on the choice of parameters. • Implicit equation: f ( x, y, z, t ) � 0 inside V f ( x, y, z, t + ∆ t ) = f ( x 0 , y 0 , z 0 , t ) + ∇ f ( M 0 ) � ∆ M + ∂f ∂t ∆ t + · · · � � � � � � � � � � � � � � � � � � � � � � � � � � � • Geometrical displacement velocity (intrinsic, � � scalar): ∂f ∆ M ∂t v S � n = lim ∆ t = − |∇ f | ∆ t → 0 Two-phase flow balance equations 2/39
� � � LEIBNIZ RULE • 3D-extension of the derivation of integrals theorem: � � � � � � � � � � d ∂f f v S � n d S f d V = ∂t d V + d t V ( t ) V ( t ) S ( t ) • Differential geometry theorem, S arbitrary. • n points outwardly (always). � � � � • Use: commutes time derivative and space integration. � � � � • Material control volumes → arbitrary volumes. Two-phase flow balance equations 3/39
� � � GAUSS THEOREM • Divergence is the flux per unit volume: � 1 ∇ � B � lim n � B d S V ǫ ǫ → 0 S • Divergence theorem, Gauss-Otstrodradski (Green) : � � ∇ � B d V = n � B d S V ( t ) S ( t ) • Differential geometry theorem, S and V arbitrary, n et ∇ on the same side. n , points outwards. B , arbitrary tensor. • Use: some particular volume integrals ⇔ surface integrals. Two-phase flow balance equations 4/39
MATERIAL VOLUMES-ARBITRARY VOLUMES • Let V m ( t ), limited by S m ( t ) be a material volume : v S m � n = v � n . � � � d ∂f f v � n d S f d V = ∂t d V + d t V m ( t ) V m ( t ) S m ( t ) • Consider V ( t ) which coincides with V m ( t ) at t . � � � d ∂f f v S � n d S f d V = ∂t d V + d t V ( t ) V ( t ) S ( t ) • Identity: for all V ( t ) which coincides with V m ( t ) at time t , � � � d f d V = d f ( v − v S ) � n d S f d V + d t d t V m ( t ) V ( t ) S ( t ) Two-phase flow balance equations 5/39
A SIMPLE EXAMPLE: MASS BALANCE • Principle: the mass of a material volume is constant. � d ρ d V = 0 d t V m ( t ) • Use the identity with f = ρ , � � d ρ ( v − v S ) � n d S ρ d V + = 0 d t V ( t ) S ( t ) � �� � � �� � Mass of V, m Net mass flux leaving S, M • The time variation of the mass of V , m , equals the net incoming mass rate, − M . d m d m d t + M = 0 , d t = − M • First principles can be formulated on material or arbitrary volumes. Both statements are equivalent . Two-phase flow balance equations 6/39
MASS BALANCE • The time variation of the mass equals the net mass flow rate entering in the volume V ( ∀ V ). � � d ρ ( v − v S ) � n d S. ρ d V = − (1) d t V S • Particular cases, – For a fixed volume, v S � n = 0, – For a material volume, v S � n = v � n Two-phase flow balance equations 7/39
SPECIES BALANCE • The time variation of the mass of component α equals (i) the net incoming mass rate of α and (ii) the production in the volume V ( ∀ V ). � � � d ρ α ( v α − v S ) � n d V + ρ α d V = − r α d V d t V S V • Add all equations for α gives the mixture mass balance. • � α r α = 0. • Chemicals redistribution, no overall net mass production. Two-phase flow balance equations 8/39
LINEAR MOMENTUM BALANCE • The time variation of the linear momentum equals the sum of (i) the in- coming momentum flux, (ii) the applied forces ( ∀ V ). � � � � d ρ v ( v − v S ) � n d S + n � T d S + ρ v d V = − ρ g d V (2) d t V S S V • T : stress tensor, contact forces. • g : volume forces. • NB: vector equation. Two-phase flow balance equations 9/39
ANGULAR MOMENTUM BALANCE • The time variation of the momentum moment equals the sum of (i) the net in- coming flux of moment of momentum and (ii) the applied torques ( ∀ V ). � � � � d ρ r × v ( v − v S ) � n d S + r × ( n � T ) d S + ρ r × v d V = − r × ρ g d V d t V S S V (3) • When torques results only of applied forces (non polar fluids). Take two get the third. – The stress tensor is symmetrical. – The linear momentum balance is satisfied. – The angular momentum balance is satisfied. Two-phase flow balance equations 10/39
TOTAL ENERGY BALANCE • Equivalent to the first principle of thermodynamics: the time variation of the total energy of a closed system equals the sum of (i) the thermal power added and (ii) the power of external forces applied to the system. • The time variation of the total energy (internal and mechanical) equals the sum of (i) the incoming total energy flux, (ii) the mechanical power of the applied forces and (iii) the thermal power given to the system.( ∀ V ). � � � � � � d u + 1 u + 1 2v 2 2v 2 ( v − v S ) � n d S ρ d V = − ρ d t V S � � � � q ′′′ d V ( n � T ) � v d S + ρ g � v d V − q � n d S + + S V S V (4) • q ′′′ : volume heat sources (Joulean heating, radiation absorption, etc .) NOT of thermodynamical origin, heat of reaction, phase transition of any order... • The process is arbitrary: reversible or not. Two-phase flow balance equations 11/39
ENTROPY BALANCE AND SECOND PRINCIPLE • The time variation of the entropy of a closed and isolated system is non negative. • The time variation of the entropy equals (i) the net inflow of entropy, (ii) the entropy given to the system in a reversible manner, (iii) the entropy sources ( ∀ V ). � � � � � q ′′′ d ρs ( v − v S ) � n d S − n � j s d S + ρs d V = − T d V + σ d V, d t V S S V V (5) σ � 0 . • The second principle is ”only” σ � 0. • When reversible, σ = 0. Two-phase flow balance equations 12/39
GENERALIZED BALANCE EQUATION Balance equations have similar forms, � � � � d n � ρ ( v − v S ) ψ d S − n � j ψ d S + ρψ d V = − φ ψ d V. d t V S S V Balance ψ φ ψ j ψ Mass 1 Species α ω α r α j α L. momentum − T ρ g v − T � R ( ∗ ) A. momentum r × v r × ρ g u + 1 2 v 2 q − T � v ρ g � v + q ′′′ Total energy σ + q ′′′ Entropy s j s T (*) R , R ij = ǫ ijk r k Two-phase flow balance equations 13/39
PRIMARY LOCAL EQUATIONS Leibniz rule, � � � � ∂ρψ n � ρ v ψ d S − n � j ψ d S + ∂t d V = − φ ψ d V. V S S V Gauss theorem, ∀ V ⊂ D f , � ∂ρψ � � ∂t + ∇ � ( ρ v ψ ) + ∇ � j ψ − φ ψ d V = 0 V Instantaneous local primary balances, ∂ρψ = −∇ � ( ρ v ψ ) −∇ � j ψ + φ ψ ∂t � �� � � �� � ���� Convection Diffusion Source Balance on a fixed and infinitesimal volume, strictly equivalent to first principles. Two-phase flow balance equations 14/39
TOTAL FLUX FORM Total flux form (Bird et al. , 2007), stationary flows. ∂ρψ = −∇ � j t ψ + φ ψ ∂t Balance total flux convective flux diffusive flux j t ρψ v j ψ ψ Mass n = ρ v Species n α = ρω α v j α Momentum φ = ρ vv − T � 2 v 2 � u + 1 q − T � v Total energy e = ρ v j t Entropy s = ρs v j s NB: Some authors may use different sign conventions for fluxes. Don’t pick up an equation from a text without care... Two-phase flow balance equations 15/39
CONVECTIVE FORM Combine with the mass balance, ∂ρ ∂t = −∇ � ( ρ v ) Expand products in the balance equation, ∂ρψ = −∇ � ( ρ v ψ ) − ∇ � j ψ + φ ψ ∂t ρ∂ψ ∂t + ψ ∂ρ ∂t = − ψ ∇ � ( ρ v ) − ρ v � ∇ ψ − ∇ � j ψ + φ ψ Definition of the convective derivative: D f D t = ∂f ∂t + v � ∇ f ρ D ψ D t = −∇ � j ψ + φ ψ Balance on a material volume (infinitesimal). Only diffusive fluxes. Practical form to derive secondary equations. Two-phase flow balance equations 16/39
SUMMARY OF CONTINUUM MECHANICS EQUATIONS For a pure fluid, on an arbitrary control volume, • Mass balance (1) • Linear momentum balance (2) • Angular momentum balance (3) • Total energy balance (4) • Entropy inequality (5) Local primary balance equations, (1) → Mass balance (6) (2) → Momentum balance (7) (3) → Stress tensor symmetry (4) → Total energy balance (8) (5) → Entropy inequality (9) Two-phase flow balance equations 17/39
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