= Unit outward normal to the interface contained within the cross section ( to

Two-Phase Flow Equations n z c a n n c Note: Walls are drawn such that n z = Unit vector in the flow direction n = Unit outward normal vector to the phase surface A x = constant ut this doesn t have to hold true n c = Unit outward normal to the interface contained within the cross section ( to z n ) 1

Transverse View a = Cross sectional area of phase c = Contour etween phases i c = Contour etween phase and the wall Limitin forms of Leinitz Rule F dc 1) FrtdS (, ) = ds Fv i n t t n n and Gauss s Theorem a a c c ) dc BdS = B n ds B n z n n z a a c i c c c Similar to that shown for sinle phase flow a n n B = n = = dc z 3) z z c n nc Bein with the eneral alance equation written for phase 4) ( v ) t ψ ψ = ψ ψ s Interate the eneral alance equation over the area a 5) ds ( v) ds sds ds t ψ ψ = ψ ψ a a a a and apply Gauss s Theorem and Leinitz Rule

6) dc ψds ψv nzds ψ( v vi) n = t z n n a a ci c dc ψ nds ψ n ψ ds z n n s z s c a ci c ac c Note: ( v v dc i) n ψ = 0 such that n n c c 7) dc ψds ψv nzds ψ( v vi) n = t z n n a a ci dc ψ nds ψ n ψ ds z n n s z s c a ci c ac c Mass Conservation ψ = ψ s = 0 ψ = 0 dc 1) ds v nzds ( v vi) n = 0 t z n n a a ci c Also a a1 a a Under conditions such as su-cooled nucleate oilin, we could have a situation such that at some interfaces we would have evaporation and at others condensation. To model such a situation, we would need mass, enery and momentum equations for each species of a phase, as well as terms to descrie how one species transitioned to the other, i.e. a 1 a 3

Define Γ ( v vi) n which represents mass exchane at the interface due to evaporation and condensation dc ) ds v nzds Γ = 0 t z n n a a ci In terms of our averain notation dc 3) < > a < vz > a Γ = 0 t z n n Let a =< α > A x ci c c δ ci Γ dc n n c 4) Ax < > < α > < v > < α > Ax δ = 0 t z Momentum Conservation ψ v ψ ψ = = s = T = ( PI σ ) 5) n vds vv nds z v( v vi) dc= t z c n i nc a a n n PI nzds PI dc σ nzds σ dc ds z n n z n n c c a ci c a ci c a Proect alon the flow direction y dottin aainst the unit vector n z 6) n vds vvds v( v vi) dc= t z c n i nc a a n nz n n PdS P dc σzzds σ dc zds z n n z n n c c a ci c a ci c a 4

7) Let: ci c c ci dc Ax < v > < α > < vv > < α > Ax Γ v = t z n nc ci nz n < α > Ax P dc < σzz >< α > Ax z n nc z c n n n n σ dc σ dc< > A < α > c z z x z n nc n nc ci Γ dc v vˆ n n =δ c n n n n a P dc = dc = = < α > A z z 1 1 1 x n nc n nc z z c n n σ dc = < τ > P z w 1 w n nc n n σ dc =< τ > P z i 1 i n nc where P i is the interfacial perimeter. Sustitutin into the area averaed momentum equation ives A ˆ x < v > < α > < vv > < α > Ax δ v = t z < α > A < α > A < σ > < α > A z z z < τ > P < τ > P< > A < α > 8) { } x 1 x zz x w 1 w i 1 i x z Internal Enery 9) ( u ) ( uv) = P v q t Aain, the internal enery equation does not satisfy the eneral alance equation, ut we can still employ our area averain techniques. Interate over the area a, applyin Leinitz Rule and Gauss s Theorem 5

n 10) uds ( uv ) nds z ( u)( v vi) dc t z n nc a a ci = P v ds q nzds n n q dc q dc z n n n n a c c a c ci 11) dc Ax < α > < uv > < α > Ax Γ v = t z n n P v ds q P q P a w w i i ci c Let a P v ds = v ds a Note, the weihtin is different thus the pressure is different than the other area averaed pressure. n =< > =< > dc n = v nzds ( v vi) n vi dc z n n c n n c a c c d dc n = < v > < α > Ax Γ vi dc dz n n n n P vds P vds P v nzds v dc z n nc a a a c ci c c c Recall Leinitz Rule vi n FrtdS (, ) FrtdS (, ) Frt (, ) dc t = t n n a a c v n dc = ds ds i n nc t t c a a = a a t t c a t = 6

Such that the internal enery equation is 1) A u α u v α A u x < >< > < > < > x Γ = t z n nc c < v > < α > Ax < α > Ax q wpw q ipi z t dc Jump Conditions Jump conditions provide the couplin of the phasic equations across the interface. The eneral form of the ump condition is v v n = v v n 13) [ ψ ( ) ψ ] [ ψ ( ) ψ ] a a i sa a i s Continuity (Mass) ψ = ψ = 0 s ( v v ) n = ( v v ) n a a i a i Γ = Γ a Note: na nac = n nc c dc dc Γ a = Γ n n n n δ = δ a a ac c c 7

Momentum ψ = v ψ = PI σ s v ( v v ) PI σ n = v ( v v ) PI σ n a a a i a a a i Proect alon flow direction y dottin aainst n z ava( va vi) PI a nz σa n z n a= v( v vi) PI nz σ n z n Γ v Pn n σ n n = Γ v Pn n σ n n a a a z a a z a z a z dc dc Γ v Pn n σ n n = Γ v Pn n σ n n a a a z a a z a a a z z na nac n nc dc dc dc dc δ ˆ ˆ ava Pn a z na σanz na = δ v Pn z n σnz n n n n n n n n n a ac c ac c c ci ci ci ci ci Assumin that pressure is continuous on the interface δ < > z a ˆ ava τia 1 Pi Pa n a n ac ci δ vˆ < τ > P = δ vˆ < τ > P a a ia 1 i i 1 i n n nz n dc = δ ˆ v< τi> 1 Pi P n n c i a ac dc Total Enery ( ) ( ) e ( v v ) q PI σ v n = e ( v v ) q PI σ v n a a a i a a a a a i Note v v e = u 8

a aua( va vi) va va( va vi) q a Pa( va vi) Pv a i σa va na = u ( v v ) v v ( v v ) q P( v v ) Pv σ v n i i i i a aha( va vi) va va( va vi) q a Pv a i σa va na = h ( v v ) v v ( v v ) q Pv σ v n i i i Assume the inetic enery and viscous terms alance a v v ( v v ) σ v = v v ( v v ) σ v such that a a a i a a i h v v q Pv n [ ( ) ] a a a i a a i a = h v v q Pv n [ ( ) ] i i Multiply oth sides y dc n n and interate over the interfacial contour c dc dc dc Γ h q n Pv n = a a a a a i a na nac na nac na nac ci ci ci dc dc dc Γ h q n Pv n i n nc n nc n nc ci ci ci Assume pressure is continuous on the interface, such that ˆ dc δ aha q iapi Pv a i na n n ci a ac ˆ dc = δ h q ipi Pv i n n n ci c δ hˆ q P = δ hˆ q P a a ia i i i 9

Two Phase Equation Summary Apply the area averaed two-phase equations to a liquid vapor system. We will aain assume all averae pressures are the same, the averae of products is the product of averaes and we can drop the averae notation. Liquid Mass A x α α v Ax = δ t z Vapor Mass A α α v A = δ t z x x Liquid Internal Enery ˆ Ax αu αu v Ax δ h P αv Ax P αax q w Pw q = ipi t z z t Vapor Internal Enery ˆ Ax α u α uvax δ h = P αvax P αax q wpw q ipi t z z t Liquid Momentum A ( α v ) ( α v v ) A = vˆ A P δ α τ P τ P αa t z z x x x w w i i x z Vapor Momentum P A ( α v ) ( α v v ) A = δ vˆ α A τ P τ P α A t z z x x x w w i i x z Jump Conditions δ = δ δ hˆ q P = δ hˆ q P i i i i δ vˆ τ P = δ vˆ τ P i i i i 10

The area averaed two-phase equations constitute the Six Equation model for modelin two phase systems as it contains six conservation equations, one for each phase. Assumin the interfacial and wall interaction terms can e expressed in terms of the followin fundamental variales, we have as unnowns Fundamental Variales, u, v, P, α, T 1 Equations: Conservation Equations 6 State Equations: = ( u, P) T = T ( u, P ) Volume Constraint α = 1 1 An additional equation is require that relates the phase pressures P = P( P ) 1 which closes the system. A common assumption is that the phase pressures are equal, i.e. P = P = P which leads to the six equation, sinle pressure model that is the asis for most desin codes. Liquid Mass A x α α v Ax = δ t z Vapor Mass A α α v A = δ t z x x Liquid Internal Enery ˆ Ax αu αuvax δ h P αvax P αax q w Pw qp = i i t z z t 11

Vapor Internal Enery ˆ Ax α u α uvax δ h = P αvax P αax q wpw q ipi t z z t Liquid Momentum A ( α v ) ( α v v ) A = δ vˆ α A P τ P τ P αa t z z x x x w w i i x z Vapor Momentum A ( α v ) ( α v v ) A = δ vˆ α A P τ P τ P α A t z z x x x w w i i x z Jump Conditions δ = δ δ hˆ q P = δ hˆ q P i i i i δ vˆ τ P = δ vˆ τ P i i i i 1

Mixture Equations The Mixture Equations are a convenient form of the two-phase equations which are otained y addin the phasic equations. The Mixture Equations provide loal alances on mass, enery and momentum as opposed to alances on each phase. Mixture Mass Ax ( α α ) ( αv α v) Ax δ δ = 0 t z 0 Mixture Momentum P A ( α v αv ) ( αv v α vv) A = A τ P τ P ( α α) A t z z δ vˆ τ ˆ i Pi δ v τipi x x x w w wl wl x z 0 Mixture Internal Enery A ( α u α u ) ( α u v α u v ) A P ( α v α v ) A q = P t z z q P δ hˆ q ˆ i Pi δ h q ipi x x x w w wl wl 0 Mixture Variale Definitions α α v αv αv = G u αu αu v v τ P τ P τ P w w w w wl wl qp w w = q wpw q w Pw The mixture equations then simplify to 13

Mixture Mass Ax vax = 0 t z Mixture Momentum A v ( α v v α v v ) A A P = τ P A t z z x x x w w x z Mixture Internal Enery A u ( α u v α u v ) A P ( α v α v ) A q = P t z z x x x w w The convective terms in the momentum and internal enery equations are all of the form αvφ αvφ. We wish to express these convective terms as mixture quantities plus correction terms, i.e. α vφ α v φ = φ v Correction Terms Define φ α φ α φ Note: This is already consistent with our definitions of u and v aove. φ v= v α φ α φ v φ v = α φ αφ αv αv φ v = [ αφ αφ] α α φ v = [ αφv αφ v ] [ αφv αφ v] α α φ v = [ α φ v α φv α φ( v v )] [ α φ v α φv α φ ( v v )] 14

φ v = ( α α ) αα αα [ αφv αφ ] ( ) v φ v v φ( v v ) αα φ v= αφv α φv ( φ φ)( v v ) Internal Enery Equation a) Convective Enery Term αvφ αvφ = αvu αvu φ = u φ = u φ = u : ) Pressure Wor Term αvφ αvφ = αv αv φ = υ φ = υ φ = υ : d αα A ( u) ( uva ) = P ( va ) q P ( u u)( v v) A t z dz z x x x w w x αα P ( υ υ )( v v) A z x Momentum Equation Momentum Flux Term αvφ αvφ = αvv αvv φ = v φ = v φ = v : P αα A ( v) ( vva ) = A τ P A ( v v ) A t z z z x x x w w x z x The Mixture Equations are not linearly independent from the phasic equations, and as such can e used to replace a phasic equation, ut can not e used in addition to the phasic equations. Since the mixture equations reduce to the sinle phase equations for α = 0, they are convenient for handlin phase appearance and disappearance. Cominations of Mixture Equations and phasic equations can e produce various simplifications to the Six Equation Models. 15

5 Equation Models Five Equation Models are composed of 3 mixture and phasic equations. Two common Five Equation models are: a) Phasic Equations = Mass Momentum Since we only have the mixture enery equation, information reardin enery distriution amon the phases has een lost. Assume least massive phase to e at saturation (could e liquid or vapor u ( P, T) = u ( P) and ( Pu, ) = ( P) s ) Phasic Equations = Mass Enery s Since we have only the mixture momentum equation, information reardin relative phase velocity has een lost. The eneral approach it to correlate the relative velocity in terms of the other system variales, i.e. vr = v v = vr( α,, u, P) Correlation Drift Flux Models are a common example of this approach. Four Equation Models Four Equation Models are composed of three mixture equations and one phasic equation. The most common of the four equation models consist of the three mixture equations and the liquid phase mass equation. The liquid phase mass equation is A x α α v Ax = δ t z Since liquid phase velocity alone does not appear in the mixture equations, we can eliminate liquid phase velocity in favor of mixture velocity and relative velocity y notin v α v v = = α v α v α v α v α v v = v = ( α α ) v α ( v v ) 16

v v α = ( v v ) Sustitutin into the liquid phase mass equation ives αα A α αva = δ ( v v) A t z z x x x which is of the same eneral form as the mixture equations. As we have only the mixture momentum equation, a correlation is required for relative velocity. In addition, since we have only the mixture enery equation, the vapor phase is usually taen to e at saturation. The four equation model does allow for the liquid phase to e sucooled. Three Equation Models Three equation models are ased solely on the mixture equations. A correlation is required for relative velocity and when two phases are present, they are oth assumed to e at saturation. In either the 3, 4 or 5 equation models homoeneous flow is otained y simply settin v r = 0. 17

Drift Flux Models The Drift Flux is defined in terms of the relative velocity as We can also define a Drift Velocity such that = α α ( v v ) = α α v r V = α ( v v ) = α v r For ravity dominated flows in the asence of wall shear, an equation which has een found to correlate a wide variety of data ives n = αα v where v is the terminal rise velocity of a sinle ule in an infinite fluid. The relative velocity, Drift Velocity and Drift Flux are then all proportional to the terminal rise velocity. One model for terminal rise velocity can e otained from the followin force alances. Buoyancy Force: V ( ) Surface Tension: Dra Force: Pσ C A d v Balancin the Buoyancy and Surface Tension forces ives V ( ) = Pσ V P = σ d ( ) ˆ σ ( ) dˆ Balancin the uoyancy and dra forces ives v V( ) = CdA 18

19 and solvin for v 1 ˆ ( ) d Cd V v C A = ( ) ( ) ( ) d d v C C C C σ σ = = 1/4 ( ) v C σ =

Numerical Solution of the Three Equation Model Numerical solution of the two phase equations is ased on the assumption of Gloal Compressiility, i.e. the spatial pressure distriution is unimportant and system parameters can e evaluated at the loal system pressure. This eliminates the need for a spatially discretized momentum equation. Examine the open channel illustrated aove. Fluid enters sucooled and can leave sucooled, a two phase mixture or superheated. We assume the 3 equation model is valid where the relative velocity vr = v v is availale y an appropriate correlation. Consistent with the three equation model, oth phases are at equilirium when two phases are present. The oundary conditions are inlet velocity, pressure, density and internal enery. We also assume a now exit pressure. Equations: Mixture Mass A x x = t z ( uva ) 0 Mixture Internal Enery d αα A ( u) ( uva ) = P ( va ) q P ( u u)( v v) A t z dz z αα P ( υ υ )( v v) Ax z x x x w w x 0

Interate the mass and enery equations over a node centered at and ounded y ± 1/ Mass d V ( vax) 1/ ( vax) 1/ 0 dt = Internal Enery d V ( u) ( uvax) 1/ ( uvax) 1/ = P[( vax) 1/ ( vax) 1/ ] q dt αα αα υ υ 1/ 1/ ( u u)( v v) Ax P ( )( v v) Ax 1/ 1/ S The term S contains the two phase correction terms and is zero for sinle phase flow. Semi-Implicit Time Discretization W assume a Semi-Implicit time discretization, where velocities are evaluated at new time, and convected properties are evaluated at old time Mass V v A v A Δt tδt t t tδ t t tδt 1/ 1/ 1/ 1/ 1/ 1/ = 0 Internal Enery ( u) ( u) V u v A u v A P v A v A q S Δt tδt t t tδ t t tδ t t tδ t tδt t t [( ) x] 1/ [( ) x] 1/ = 1/ x 1/ 1/ x 1/ Consistent with our treatment of convected properties in the sinle phase equations, we assume any oundary valued property Ψ can e represented in terms of the cell centered properties y v Ψ = Ψ Ψ Ψ Ψ tδt t 1 t t 1 t t 1/ 1 tδt 1 v 1/ 1/ { } { } 1

State Equations a) Sinle Phase = ( u, P) ( u) = ( u, P) u ) Two Phase = α α = ( P) α [ ( P) ( P)] f f ( u) = αu αu = f ( Pu ) f ( P) α f ( Pu ) ( P) f ( Pu ) f ( P) These state equations can e expressed in eneral as = ψ (, ) P ( u) = u( ψ, P) For Sinle Phase ψ = u and Two Phase ψ = α The Mass and Internal Enery Equations are linear in the new time values. The equations are nonlinear as a result of the state equations. Linearize the equations via the Newton-Raphson technique Mass V v A v A Δt 1 t t 1 t 1 1/ 1/ 1/ 1/ 1/ 1/ = 0 Internal Enery V u v A u v A P v A v A q S Δt 1 t ( u) ( u) t 1 t 1 t 1 1 t t [( ) x] 1/ [( ) x] 1/ = 1/ x 1/ 1/ x 1/

State = ( ψ, P ) ( ψ ψ ) ( P P ) P 1 1 1 ψ u u u = u( ψ, P ) ( ψ ψ ) ( P P ) P 1 1 1 ψ u where the state equation derivatives are a) Sinle Phase = ( up, ) ψ = u = ψ u P = P P u u = ( u, P) u u = ( up, ) u ψ u P u P = u P u ) Two Phase ( α, P) = ( P) α [ ( P) ( P)] ψ α f f = ψ = f f = α α P P P 3

u( α, P) = ( P) u ( P) α [ ( P) u ( P) ( P) u ( P)] f f f f u = ( Pu ) ( P) f ( Pu ) f ( P) ψ u f uf u = α uf f α u P P P P P The linearized equations can e written in the form Mass 1 t 1 t 1 t a1v 1/ a v 1/ = S1 Internal Enery ( ) 1 1 1 t t t 1 1/ 1/ = u v v S State i) ii) 1 = δψ δp ψ P 1 ( ) u u u = u δψ δp ψ P Sustitute i) into the Mass Equation, and ii) into the Internal Enery Equation p t 1 t 1 t P a1v 1/ a v 1/ = S1 ψ P δψ δ u u t 1 t 1 t δψ δp 1v 1/ v 1/ = S u ψ P Divide the Mass Equation y eliminateδψ, divide the Internal Enery Equation y ψ u ψ and sutract to 4

iii) av v cδ P= ξ 1 1 1/ 1/ For J nodes, the numer of unnowns are J velocities for inlet velocity nown and P (or δ P ). We have the J equations implied y iii) aove. One more equation is required. We assume a simplified Momentum Boundary Condition of the form ( v ) P Pexit K Δ = t t t J 1/ J 1/ which can e linearized to ive P P K ( v v ) v t 1 J 1/ 1 = exit J 1/ J 1/ J 1/ δ P= P P = dv C 1 1 J 1/ J The linear system of equations may e written in matrix form, where the matrix structure is illustrated elow where vn = v n 1/ a1 c1 v1 ξ1 v1/ a c v ξ 3 a3 c 3 v ξ 3 3 = d 1 δ P C J While not tridiaonal, solution alorithms can e developed for this structure that are similar to the Thomas Alorithm for tridiaonal systems. 5

Recirculatin Systems The flow paths within U-Tue steam enerators and Boilin Water Reactors may e approximated as the recirculatin system illustrated elow. For the one dimensional sements contained etween the inlet manifold (node 1) and the exit manifold (node J) the mass and internal enery equations are of the same form as the previous open channel prolem av v δ P= ξ [, J 1] 1/ 1/ For the exit manifold, the mass and internal enery equations reduce to av v cv δ P= ξ J exit J 1 J J and for the inlet manifold (node 1) av v cv δ P= ξ 3 1 In addition, we have the momentum oundary condition at the steam outlet d v δ P= C ex exit J Assumin the velocity v is nown, the unnowns are: v 3, v ( [, J 1] ), v exit, v, P δ J 6

The numer of equation is J1, the last equation is an interated momentum equation around the entire loop. The mixture momentum equation (valid for oth sinle and two phase conditions) is 1 P f v v v [ vva ] = φ δ( z z ) ψ t A z z D x or in non conservative form x 0 e 1 αα sin θ ( v v ) Ax Ax z v v P f v v v = φ δ( z z ψ t z z D e 0 ) 1 αα sin θ ( ) v v Ax Ax z z (interate from center of one node to center of the next) z 1 Results in an equation for the new iterate values of velocity of the form B v 1/ = e which closes the system of equations. 7