Introduction to nomenclature. Two-phase flow regimes in vertical tubes. Forced convective boiling: Regions of boiling and flow

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1 NTEC Module: Water Reactor Performance and Safety Two-phase flow regimes in vertical tubes ecture 5: Introduction to two-phase flow. F. ewitt Imperial College ondon Bubble Flow Slug or Plug Flow Churn Flow nnular Flow Wispy nnular Flow Forced convective boiling: Regions of boiling and flow 3 Introduction to nomenclature New international nomenclature (i) Capital letters: M, mass V, volume Q, quantity of heat (ii) Dotted variables: M, mass rate of flow V, volume rate of flow (iii) ower case variables: h, enthalpy c p, specific heat (iv) ower case dotted variable: q, heat flux (per unit area) m, mass flux Specific (per unit mass) 4

2 Definitions of quantities gas Total mass rate of flow = (as M, iquid M ) liquid Total mass rate of flow = (as V, iquid V ) Total cross M section area = Mass flux = m Fraction of cross section occupied by gas phase ( void fraction ) = ε Fraction occupied by liquid phase (holdup) = ε =(- ε ) Superficial velocity : V V V U, U, U U U Velocities: u, u Slip ratio: u S u M V 5 Basic equations: The homogeneous model: Introduction Void fraction: V x V x ( x) / omogeneous flow model treats fluid as single fluid cf. single phase flow assumes equal velocities for two phases omogeneous density M m V mx m ( x) x ( x) Slip ratio: S u/ u 6 Basic equations: omogeneous continuity (mass) equation For channel element (Slide 5) Rate of creation Mass outflow = 0 = of mass rate - Mass inflow rate z Mass storage + rate t 0 U z U U z U = total volume flux Rearranging gives: z U 0 t 7 Basic equations: omogeneous momentum equation Rate of creation Momentum of momentum = outflow rate - Momentum Momentum Inflow rate + storage rate mu z mu mu m z z t sum of forces on element sum of forces to pressure gradient, gravity and wall shear p p p z gz sin ozp z p fluid pressure (assumed constant over cross section); m P channel periphery; U Rearranging gives homogeneous momentum equation: m m / p g sin t z z 8

3 Rate of creation of energy Basic equations: omogeneous energy equation = 0 = Energy outflow rate - Energy Inflow rate + Energy storage rate U 0 Ue z Ue U e qp qv z z z t U eenergy convected per unit mass of fluid h gzsin p h enthalpy internal energy of fluid q wall heat flux q v volumetric heat generation rate Rearranging gives homogeneous energy equation: e e qp p U q v 9 t z t omogeneous model: Components of pressure gradient omogeneous momentum equation (Slide 7) 8) o m m / p g sin t z z For constant cross section and steady flow p P d m z dz g sin Total Frictional cceleration ravitational dp dp dp dpg dz dz dz dz F a 0 Frictional pressure drop calculation: omogeneous model From slide 9 Define f ence dp P P D 4 f F o dz D /4 D TP TP for round tubes from o / m / dpf ftpm, dz D x ( x) ftp fnre TP from standard curves md x x Re TP, (Mcdams et al., 94) TP TP (usually very poor predictions) Mcdams et al., TSME 64, 93 (94) Basic equations: Separated flow model: Introduction u u q Two fluids flow in separate regions each with a given but unequal velocity: Thus: u u, S Mean fluid density in element: TP Void fraction / slip ratio relationship: x x S x / x quality (fraction of mass flow flowing as vapour) 3

4 Basic equations: Balance element for Separated flow model Separated flow model (six equation model). Main features: Two flow regions Continuity, momentum and (energy) balances written Equations written for each phase Equations may be summed to give overall balances. 3 Conservation equations: Continuity equation for the liquid phase Rate of creation of mass = 0 = mass outflow rate mass in flow rate + mass storage rate u z u m ez (outflow) z u (inflow) z t 0 (storage) u m = void fraction = liquid phase velocity = cross sectional area = rate of conversion of e liquid to vapour per unit length 4 Conservation equations: Continuity equation for the gas phase Rate of creation of mass = 0 = mass outflow rate mass in flow rate + mass storage rate z um ez z t 0 u z u (inflow) (storage) (outflow) = void fraction u m e t z = liquid phase velocity dding the equations and noting that: = cross sectional area m u u = rate of conversion of We have: TP e liquid to vapour per unit length m 0 5 TP 6 t z u m Conservation equations: Continuity equation: iquid, gas, combined Simplifying from previous slide gives liquid phase continuity equation: u me t z Similarly for gas phase: 4

5 Conservation equations: Momentum equation for liquid phase Rate of creation Momentum of momentum = outflow rate M u z M u M u u z z t z u u z t (since M u ) = - Momentum Momentum Inflow rate + storage rate Sum of forces acting on control volume (next slide) Momentum equation for liquid phase: Forces on element of fluid p p z p z (Net pressure force on ends of element) pz z (Pressure force on resolved area of sloping liquid surface) g zsin Pz Pz o i i (gravitational force) (shear forces) o i = wall shear stress = interfacial shear stress 7 8 Momentum equation for liquid phase: Equating momentum creation with forces p 0P ipi ( ) g( ) sin z u( ) u( ) t z Steady state, constant area duct dp 0P ipi ( ) g( ) sin dz d u( ) dz 9 Momentum equation for gas phase: (Similar derivation to that for liquid) p ipi g sin z ( u ) ( u ) t z Steady state, constant area duct: p ipi g sin z ( u z ) 0 5

6 Mixture momentum equation dding the liquid (slide 9) and gas (slide 0) momentum equations we have: p gtp sin z u u t u u z TP Putting: Mixture momentum equation: lternative form m u u u m x u mx ives mixture momentum equation in form: x p 0P m x gtp sin m z t z For steady state flow in a duct of constant : x x p m gtp sin z z Combined conservation equations for separated flows Continuity: TP m 0 t z Momentum: p gtp sin z m x x m t z Energy: h h m xh xh t z m x x p qp qv z t TP - 3 Correlations for steady state channel flows (constant cross section) OMOENEOUS MODE correlation dp / d m g sin dz dz dp dpf dp dp a g dz dz dz dz Friction cceleration ravity dp d x x m gtp sin dz dz SEPRTED FOW MODE and correlation o o 4 6

7 Frictional pressure gradient Frictional pressure gradient in a round tube: Pressure drop multipliers: ravitational pressure gradient ravitational component of pressure gradient given by: Definition of slip ratio: Void fraction/slip ratio relationship: Void fraction in homogeneous flow: 5 6 Typical correlation for void fraction (Premoli et al, 97) Typical correlation for friction: Freidel,

8 Distribution of errors for Combination: Premoli et al (97)/Freidel (979) Pressure drop in singularities: Typical types of singularity p m p p m m m p p m p Expect large errors with empirical correlations 9 (a) (b) (c) 30 Pressure drop in singularities: omogeneous model for singular pressure drops Sudden enlargement m 90 bend p p m Sudden contraction p p kb k 0.5 m p p omogeneous multiplier Cc x, C c contraction coefficient Cc (Chisholm) / B Stages: Phenomenological approach: eneral principles () Identify the type of interfacial distribution i.e. FOW REIME () Observe detailed phenomena and make appropriate measurements. (3) Construct physical models of theoretical or semitheoretical type to describe the phenomena. (4) Integrate the local models to achieve a complete system description. 3 8

9 Phenomenological approach: Example: nnular flow I Phenomenological approach: Example: nnular flow II Phenomenological approach: Example: nnular flow III Phenomenological approach: Example: nnular flow IV

10 Phenomenological approach: Example: nnular flow V Phenomenological approach: nnular flow VI Correlation for deposition rate D = kc See ewitt, Invited ecture, SME 990, Winter nnual Meeting, Dallas Phenomenological approach: nnular flow VII Correlation for entrainment rate Phenomenological approach: Example: nnular flow V m FC critical film flow for onset of entrainment 39 Prediction of Nigmatulin (978) high pressure film flow data (ewitt & ovan, 990) 40 0