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
[email protected], 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 conditions). • 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
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MATHEMATICAL TOOLS v = v S
0
M ( u 0,v 0) u = u 0
• Displacement velocity of a surface, S: ∂M vS , ∂t u,v • Depends on the choice of parameters. • Implicit equation: f (x, y, z, t) 6 0 inside V f (x, y, z, t + ∆t) = f (x0 , y0 , z0 , t)
n f ( x ,y ,z ,t+ , t) = 0 f ( x ,y ,z ,t) = 0 , M +
M 0
∂f ∆t + · · · +∇f (M0 ) ∆M + ∂t • Geometrical displacement velocity (intrinsic, scalar):
-
∂f ∆M vS n = lim = − ∂t ∆t→0 ∆t |∇f | Two-phase flow balance equations
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LEIBNIZ RULE
v
v 5.n n 5
n
• 3D-extension of the derivation of integrals theorem: Z Z Z d ∂f dV + f dV = f vS n dS dt V (t) ∂t V (t) S(t) • Differential geometry theorem, S arbitrary. • n points outwardly (always).
8 J 5 J
• Use: commutes time derivative and space integration. • Material control volumes → arbitrary volumes.
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GAUSS THEOREM • Divergence is the flux per unit volume: Z 1 ∇ B , lim n B dS →0 V S n
5
8
• Divergence theorem, Gauss-Otstrodradski (Green) : Z Z ∇ B dV = n B dS 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.
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MATERIAL VOLUMES-ARBITRARY VOLUMES • Let Vm (t), limited by Sm (t) be a material volume : vSm n = v n. Z Z Z d ∂f dV + f v n dS f dV = dt Vm (t) Vm (t) ∂t Sm (t) • Consider V (t) which coincides with Vm (t) at t. Z Z Z ∂f d dV + f vS n dS f dV = dt V (t) S(t) V (t) ∂t • Identity: for all V (t) which coincides with Vm (t) at time t, d dt
Z
f dV =
Vm (t)
d dt
Z V (t)
f dV +
Z
f (v − vS ) n dS
S(t)
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A SIMPLE EXAMPLE: MASS BALANCE • Principle: the mass of a material volume is constant. Z d ρ dV = 0 dt Vm (t) • Use the identity with f = ρ, Z Z d ρ dV + ρ(v − vS ) n dS = 0 dt V (t) S(t) | {z } | {z } 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 . dm + M = 0, dt
dm = −M dt
• First principles can be formulated on material or arbitrary volumes. Both statements are equivalent.
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MASS BALANCE • The time variation of the mass equals the net mass flow rate entering in the volume V (∀V ). Z Z d ρ dV = − ρ(v − vS ) n dS. (1) dt V S • Particular cases, – For a fixed volume, vS n = 0, – For a material volume, vS n = v n
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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 ). Z Z Z d ρα dV = − ρα (vα − vS ) n dV + rα dV dt V S V • Add all equations for α gives the mixture mass balance. P • α rα = 0. • Chemicals redistribution, no overall net mass production.
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LINEAR MOMENTUM BALANCE • The time variation of the linear momentum equals the sum of (i) the incoming momentum flux, (ii) the applied forces (∀V ). Z Z Z Z d ρv dV = − ρv(v − vS ) n dS + n T dS + ρg dV (2) dt V V S S • T: stress tensor, contact forces. • g: volume forces. • NB: vector equation.
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ANGULAR MOMENTUM BALANCE • The time variation of the momentum moment equals the sum of (i) the net incoming flux of moment of momentum and (ii) the applied torques (∀V ). Z Z Z Z d ρr × v dV = − ρr × v(v − vS ) n dS + r × (n T) dS + r × ρg dV dt 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.
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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 ). Z Z d 1 2 1 2 ρ u+ v dV = − ρ u + v (v − vS ) n dS dt V 2 2 S Z Z Z Z + (n T) v dS + ρg v dV − q n dS + q 000 dV S
V
S
V
(4) • q 000 : 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
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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 ). Z Z Z Z Z 000 d q ρs dV = − ρs(v − vS ) n dS − n js dS + dV + σ dV, dt V T S S V V (5) σ > 0. • The second principle is ”only” σ > 0. • When reversible, σ = 0.
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GENERALIZED BALANCE EQUATION Balance equations have similar forms, Z Z Z Z d ρψ dV = − n ρ(v − vS )ψ dS − n jψ dS + φψ dV. dt V S S V
Balance
ψ
jψ
φψ
Mass
1
Species α
ωα
jα
rα
L. momentum
v
−T
ρg
A. momentum
r×v
−T R(∗)
r × ρg
Total energy
u + 12 v2
q−Tv
ρg v + q 000
Entropy
s
js
σ+
q 000 T
(*)R, Rij = ijk rk Two-phase flow balance equations
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PRIMARY LOCAL EQUATIONS Leibniz rule, Z V
∂ρψ dV = − ∂t
Z
n ρvψ dS −
S
Z
n jψ dS +
S
Z
φψ dV.
V
Gauss theorem, ∀V ⊂ Df , Z ∂ρψ + ∇ (ρvψ) + ∇ jψ − φψ dV = 0 ∂t V Instantaneous local primary balances, ∂ρψ = −∇ (ρvψ) −∇ jψ +φψ | {z } | {z } |{z} ∂t Convection
Diffusion Source
Balance on a fixed and infinitesimal volume, strictly equivalent to first principles.
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TOTAL FLUX FORM Total flux form (Bird et al. , 2007), stationary flows. ∂ρψ = −∇ jtψ + φψ ∂t
Balance
total flux
convective flux
diffusive flux
jtψ
ρψv
jψ
Mass
n=
ρv
Species
nα =
ρωα v
jα
Momentum
φ=
ρvv
−T
Total energy
e=
ρv u +
Entropy
jts =
ρsv
1 2 2v
q−Tv js
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
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CONVECTIVE FORM Combine with the mass balance, ∂ρ = −∇ (ρv) ∂t Expand products in the balance equation, ∂ρψ = −∇ (ρvψ) − ∇ jψ + φψ ∂t ∂ψ ∂ρ ρ +ψ = −ψ∇ (ρv) − ρv ∇ψ − ∇ jψ + φψ ∂t ∂t Definition of the convective derivative: Df ∂f = + v ∇f Dt ∂t ρ
Dψ = −∇ jψ + φψ Dt
Balance on a material volume (infinitesimal). Only diffusive fluxes. Practical form to derive secondary equations. Two-phase flow balance equations
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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
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CONTINUUM MECHANICS EQUATIONS Secondary balance equations, for a pure fluid, • Mechanical energy balance, (10) v momentum balance. • Internal energy balance (11), total energy balance (8)-(10). • Enthalpy balance (12). (11), h , u + p/ρ • Entropy balance (13), (11), du = T ds − pdv (Gibbs). • Comparing to entropy inequality (9), provides js and σ.
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THE CLOSURE ISSUE (I) • In balance equations, – Local variables, v, vα , p, u, etc. – Unknown fluxes, jα , T, q, js . NB: T = −pI + V – Unknown sources, rα , σ. • First principles cannot provide expressions for fluxes. The CME are not closed. • An extended interpretation of the second principle, – provides entropy sources. For a pure fluid, T σ = q ∇T + V : ∇v. – provides the thermodynamic equilibrium conditions, σ = 0, – provides constraints on possible closure to ensure return to equilibrium. Linearity assumption, transport properties, 2 T = µ(∇v + v∇) + (ζ − µ)∇ v, 3
q = −κ∇T,
µ, ζ, κ > 0
• Transport properties must be measured or modeled beyond CME scope. Two-phase flow balance equations
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TWO-PHASE LOCAL BALANCE EQUATIONS
8 J
) 1 J
n
) E J ) J
8 1 J
• Example: mass balance, V = V1 ∪ V2 , A = A1 ∪ A2 fixed. Interfaces,surface of discontinuity. Z Z d ρ dV = − ρv n dS, ∀V dt V A • Split contributions from V1 and V2 : Z Z Z Z d d ρ dV + ρ dV = − ρv n dS − ρv n dS dt V1 dt V2 A2 A1
• For V1 (t) (not fixed), Leibniz rule: Z Z Z ∂ρ1 d ρ1 dV = dV + ρ1 vAi n1 dA dt V1 V1 ∂t Ai (t) • Gauss theorem: Z Z ρv1 n1 dS = A1
∇ (ρ1 v1 ) dS −
V1
Z
ρ1 v1 n dA
Ai
• Apply the same procedure for V2 , sum up all contributions, Two-phase flow balance equations
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TWO-PHASE MASS BALANCE • Collect integral terms wrt dimension, ∀V , 2 Z X
k=1
Vk
Z ∂ρk + ∇ (ρk vk ) dV − (ρ1 (v1 − vi ) + ρ2 (v2 − vi )) dA = 0 ∂t Ai
• Local mass balance, k = 1, 2, for all points in Vk (PDE), ∂ρk + ∇ (ρk vk ) = 0 ∂t • For all points of the interface, jump condition, ρ1 (v1 − vi ) n1 + ρ2 (v2 − vi ) n2 = 0 | {z } | {z } m ˙1
m ˙2
• Jump condition is the mass balance of the interface, m ˙ k = ρk (vk − vi ) nk .
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LOCAL BALANCE EQUATIONS • Use the generalized local balance equation, same procedure, ∀V 2 Z X
k=1
Vk
−
∂ρk ψk + ∇ (ρk ψk vk ) + ∇ (jψk ) − φk dV ∂t
Z
2 X
(m ˙ k ψk + nk jψk + φi ) dA = 0
Ai k=1
• At every points of each phase, ∂ρk ψk + ∇ (ρk ψk vk ) + ∇ (jψk ) − φk = 0 ∂t • At every points of the interface, jump condition, balance of the interface. 2 X
(m ˙ k ψk + nk jψk + φi ) = 0
k=1
• φi : entropy source at the interface. Two-phase flow balance equations
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JUMP CONDITIONS • Mass balance, ρ1 (v1 − vi ) n1 + ρ2 (v2 − vi ) n2 = 0 m ˙ 1+m ˙2=0 • No phase change: m ˙ k = 0, m ˙1=m ˙2=0 • Assumption: no slip at the interface (φi = 0), (v1 − vi ) n1 = 0,
(v2 − vi ) n2 = 0
(v1 − v2 ) n1 = 0 ⇒ v1 = v2
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JUMP CONDITIONS (CT’D) • Momentum balance, m ˙ 1 v1 + m ˙ 2 v2 − n1 T1 − n2 T2 = 0 • When no viscosity, T = −pI + V, v = vt + vn , vn = n(v n), m ˙ 1 (v1n − v2n ) + (p1 − p2 )n1 = 0 vt = vt 1
2
• General case,
m ˙ 1 (v1 − v2 ) + (p1 − p2 )n1 − n1 (V1 − V2 ) = 0
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JUMP CONDITIONS (CT’D) • Particular case: 1D flow, vk (x) ⊥ interface ⊥:
m ˙ 1 (v1 − v2 ) n1 + (p1 − p2 ) − n1 (V1 − V2 ) n1 = 0
• 1D incompressible flow,
dvk dx
= 0 ⇒ Vk = 0,
m ˙ 1 (v1 − v2 ) n1 + (p1 − p2 ) = 0 • From the mass balance, definition: m ˙ k = ρk (vk − vi ) nk , 1 1 m ˙1 − = (v1 − v2 ) n1 ρ1 ρ2 • Results, pressure jump, recoil force, ρ1 − ρ2 2 p 1 − p2 = m ˙ 1. ρ1 ρ2 • p1 − p2 ∝ ρ1 − ρ2 whatever m ˙ 1. Two-phase flow balance equations
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MOMENTUM BALANCE AND SURFACE TENSION n
+ J 8 J
) 1 J N n
2
C(t)
) E J 8 1 J
• Momentum balance on a fixed volume. Forces: Z Z Z = nk Tk dS + σN dl + ρk Fk dV.
) J
A1 ∪A2
V1 ∪V2
• Th´eor`eme de Gauss Aris (1962), Delhaye (1974) : Z Z σN dl = (∇S σ − nσ∇S n) dS C(t)
Ai (t)
• ∇S : surface gradient, ∇S : surface divergence. Momentum balance interface: m ˙ 1 v1 + m ˙ 2 v2 − n1 T1 − n2 T2 = −∇S σ + nσ∇S n • ∇S σ: Marangoni force, nσ∇S n: capillary pressure, Laplace pressure jump. nσ∇S n = 2Hn • H: mean curvature of the surface. Circular cylinder: 1/R, sphere: 2/R. Two-phase flow balance equations
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EXAMPLE: 2D INTERFACES • Momentum jump at the interface, dσ σ m ˙ 1 v1 + m ˙ 2 v2 − n1 T1 − n2 T2 + τ− n=0 dl R • For a non viscous fluid, T = −pI + V,no phase change, σ dσ τ− n=0 n1 (p1 − p2 ) + dl R σ • Laplace relation, ⊥ : (p1 − p2 ) = n1 n R dσ • Inconsistency, // : µk = 0 ⇒ =0 dl • Marangoni effect for viscous fluids, σ(T ), σ(c), dσ −(n1 V1 + n2 V2 ) τ + =0 dl • Be careful with the parameter selection. Pressure is always higher in the concavity side (balloon). Two-phase flow balance equations
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JUMP CONDITIONS (CT’D) • Total energy balance: 1 2 1 2 ˙ 2 u2 + v2 +q1 n1 +q2 n2 −n1 T1 v1 −n2 T2 v2 = 0 m ˙ 1 u1 + v1 +m 2 2 • Enthalpy form, 3 common assumptions, – phase change is the dominant effect, – variation of mechanical energy can be neglected, – the effect of pressure and viscous stress jump can be neglected (no surface tension), m ˙ 1 h1 + m ˙ 2 h2 + q1 n1 + q2 n2 = 0 • More on the derivation, see Delhaye (1974, 2008). • Thermodynamic equilibrium condition at the interface: 1 2 1 1 n2 V2 n2 n1 V1 n1 v1t = v2t , T1 = T2 , g1 −g2 = m ˙1 − − − 2 ρ22 ρ21 ρ2 ρ1 Two-phase flow balance equations
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USE OF LOCAL EQUATIONS • First principles→ balance on arbitrary control volumes – Local phase equations, – Jump conditions at the interface (see also the Rankine-Hugoniot eqs). • Flows with simple interface configuration – Stability of a liquid film, – Growth/collapse of a vapor bubble (nucleate boiling, cavitation). • More general problems, – Tremendously large number of interfaces, non-equilibrium. – Large scale fluctuations, intermittency, engineers seek for mean values. ⇒ Space averaging of local equations (area-averaged): 1D models ⇒ Time averaging of local equations: CMFD (3D codes) ⇒ Space and time averaging, composite averaging: two-fluid 1D model, 1D codes, system codes. Two-phase flow balance equations
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AREA-AVERAGED BALANCE EQUATIONS • Area-averaging operator: < fk >2 =
1 Ak
Z
fk dA
Ak
• How to get a balance equation for a mean value? Average the local balance on Ak . Example, mass balance,
n k
n
k C
∂ρk + ∇ (ρk vk ) = 0 ∂t • Integrate on Ak , Z Z ∂ρk dA + ∇ (ρk vk ) dA = 0 Ak ∂t Ak
n
C k C
A k
k
• Limiting forms of the Leibniz rule and Gauss theorems, Z Z ∂ ∂ ρk dA + · · · + ρk wk dA + · · · = 0 ∂t Ak ∂z Ak | {z } | {z } Ak 2
Two-phase flow balance equations
Ak 2
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MATHEMATICAL TOOLS • Limiting form of the Leibniz rule, Z Z Z ∂ ∂fk dl fk dA = dA + fk vi nk ∂t Ak nk nkC Ak ∂t Ck n k
n
k C
• Limiting form of the Gauss theorem, Z Z Z ∂ dl ∇ BdA = nz BdA + nk B ∂z nk nkC Ak Ak Ck • First useful identity, B = nz Z ∂Ak dl =− nk nz ∂z nk nkC Ck
n
C k C
A k
k
(1)
• Second useful identity, B = pI Z Z Z ∂ dl pnz dA + ∇p dA = pnk ∂z nk nkC Ak Ak Ck
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(2)
AREA-AVERAGED BALANCE EQUATIONS (CT’D) • Integrate on Ak , Z Ak
∂ρk dA + ∂t
Z
∇ (ρk vk ) dA = 0
Ak
• Leibniz rule and Gauss theorem, ∂ ∂ Ak < ρk >2 + Ak < ρk wk >2 = − ∂t ∂z
Z Ck
m ˙k
dl nk nkC
• Γk : production rate of phase k [kg/s/m] per unit length of pipe. Z dl Γk = − m ˙k nk nkC Ck • No phase change: m ˙ k = 0 ⇒ Γk = 0. • Mass balance of the interface, m ˙ 1+m ˙ 2 = 0 ⇒ Γ1 + Γ2 ≡ 0. • The area-averaged mass balance is not closed. Two-phase flow balance equations
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AREA-AVERAGED BALANCE EQUATIONS • Based on the general form of the local balance equations., ∂ ∂ ∂ Ak < ρk ψk >2 + Ak < nz ρk vk ψk >2 + Ak < nz jψk >2 −Ak < φk >2 ∂t ∂z ∂z Z Z dl dl =− (m ˙ k ψk + nk jψk ) − nk jψk nk nkC nk nkC Ci Ck • δAk = Ci ∪ Ck , Ck wall fraction wetted by k, Ci , interface.
C
C
A k
i
n k
z k
n k
n
C
k C
n
i
k
Two-phase flow balance equations
C
n k
n
k C
k
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MOMENTUM BALANCE • Note on the momentum balance, vector equation, ∂ ∂ ∂ Ak < ρk vk >2 + Ak < ρk wk vk >2 − Ak < nz Tk >2 −Ak < ρk gk >2 ∂t ∂z ∂z Z Z dl dl =− (m ˙ k vk − nk Tk ) + nk Tk n n nk nkC k kC Ci Ck • Projection on nz : right dot product, wk = vk nz , stress tensor decomposition, ∂ ∂ ∂ ∂ Ak < ρk wk >2 + Ak < ρk wk2 >2 + Ak < pk >2 − Ak < nz Vk nz >2 ∂t ∂z ∂z ∂z Z Z dl dl + nk Tk nz −Ak < ρk gz >2 = − (m ˙ k wk − nk Tk nz ) nk nkC nk nkC Ci Ck • Identity (1), assume < pk >2 = pC , Z Z dl dl ∂Ak − = −pC nk nz =< pk >2 pk nk nz nk nkC nk nkC ∂z Ck ∪Ci Ck ∪Ci • Other choices are possible, introduce an excess pressure, pi , pC =< pk >2 +pi · · · Two-phase flow balance equations
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MOMENTUM BALANCE (CT’D) • Momentum balance, single-pressure, < pk >2 = pC , ∂ ∂ ∂ ∂ 2 Ak < ρk wk >2 + Ak < ρk wk >2 +Ak < pk >2 − Ak < nz Vk nz >2 ∂t ∂z ∂z ∂z Z Z dl dl −Ak < ρk gz >2 = − (m ˙ k wk − nk Vk nz ) + nk Vk nz nk nkC nk nkC Ci Ck • Transfers are dominant in the radial direction, quasi fully developed flows, ∂ ∂ ∂w Ak < nz Vk nz >2 ∝ νt →0 ∂z ∂z ∂z • Give an example of a situation where this term might not be neglected. • Closures are required: interactions at the interface and the wall (wall friction).
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TIME AVERAGED BALANCE EQUATIONS [T k] fk
+ +
• Conditional time-averaging, R Z Xk fk dt Xk fk 1 X T R = fk = fk dt = Tk [Tk ] X dt Xk k T
+ +
+
J t
t-T /2
t1
t2k T
t2k+ 1
t2n
• Plain time-average, f=
t+ T /2
1 T
Z
f dt
T
• How to get a balance equation for a mean value? Average the local balance on [Tk ]. Z Z ∂ρk dt + ∇ (ρk vk ) dt = 0 [Tk ] ∂t [Tk ] • Limiting forms of the Leibniz rule and the Gauss, theorem, Z Z ∂ ρk dt + · · · + ∇ ρk vk dt + · · · = 0 ∂t [Tk ] [T ] | {z } | k {z } Tk ρk X
Tk ρk vk X
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MATHEMATICAL TOOLS • Limiting form of the Leibniz rule (derivation of an integral wrt upper limit), Z Z X ∂fk ∂ vi nk dt = fk dt − fk ∂t [Tk ] |vi nk | [Tk ] ∂t disc.∈[T ] | {z } ±1
• Limiting form of the Gauss theorem, Z Z ∇ Bk dt = ∇ [Tk ]
X
Bk dt +
[Tk ]
disc.∈[T ]
nk Bk |vi nk |
• Time-averaged mass balance, production rate of phase k, interfacial interactions are homogenized [kg/s/m3 ], 1 ∂αk ρk X X + ∇ αk ρk vk =− ∂t T
X
disc.∈[T ]
m ˙k |vi nk |
• Time-averaged balance equations, starting point of the Reynolds decomposition, T? X ∂αk ρk ψk 1 X X X +∇ αk ρk vk ψk +∇ αk jψk −αk φk = − ∂t T
Two-phase flow balance equations
X
disc.∈[T ]
m ˙ k ψk + nk jψk |vi nk | 37/39
COMPOSITE AVERAGES: THE TWO-FLUID MODEL • Example: mass balance, space-averaged and time averaged, Z ∂ ∂ dl Ak < ρk >2 + Ak < ρk wk >2 = − m ˙k ∂t ∂z nk nkC Ck • Time averaged and space averaged, ∂ ∂ 1 X X A< | αk ρk > | 2+ A< | αk ρk wk > | 2 = −A< | ∂t ∂z T
X
disc.∈[T ]
m ˙k > | 2 |vi nk |
• LHS are identical, the RHS should also. Proof, identity on interaction terms, 1 X 1 • Local specific interfacial area, γ = T |vi nk | disc.∈[T ]
• Possible closure of interaction terms: interfacial area × mean flux, 1 T
X
disc.∈[T ]
m ˙k m ˙ ki = |vi nk | T
X
disc.∈[T ]
1 = γm ˙ ki |vi nk |
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REFERENCES Aris, R. 1962. Vectors, tensors, and the basic equations of fluid mechanics. Prentice-Hall. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. 2007. Transport phenomena. Revised second edn. John Wiley & Sons. Delhaye, J. M. 1974. Jump conditions and entropy sources in two-phase systems, local instant formulation. Int. J. Multiphase Flow, 1, 359–409. Delhaye, J.-M. 2008. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences.
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