Two-Phase Gas/Liquid Pipe Flow Ron Darby PhD, PE Professor Emeritus, Chemical Engineering Texas A&M University Types of Two-Phase Flow »Solid-Gas »Solid-Liquid »Gas-Liquid »Liquid-Liquid Gas-Liquid Flow Regimes . Homogeneous . Highly Mixed . “Pseudo Single-Phase” . High Reynolds Number . Dispersed –Many Possibilities . Horizontal Pipe Flow . Vertical Pipe Flow Horizontal Dispersed Flow Regimes Vertical Pipe Flow Regimes Horizontal Pipe Flow Regime Map 1/ 2 G L AW 2 1/ 2 WLW LWL Vertical Pipe Flow Regime Map DEFINITIONS Mass Flow Rate m , Volume Flow Rate (Q) m mLGLLGG m Q Q Mass Flux (G): m m m GGG L G AAA LG Volume Flux (J): G GL GG JJJLG m L G QQ LGV A m Volume Fraction Gas: ε Vol. Fraction Liquid: 1-ε Phase Velocity: J J V,VL G LG1 Slip Ratio (S): Mass Fraction Gas (Quality x): V m S G x G VL mmGL Mass Flow Ratio Gas/Liquid: m x GGS m 1 - x 1 - LL Density of Two-Phase Mixture: m G 1 L where x ε x S 1 x ρGL / ρ is the volume fraction of gas in the mixture Holdup (Volume Fraction Liquid): S 1 x ρGL / ρ φ 1 ε x S 1 x ρGL / ρ Slip (S) . Occurs because the gas expands and speeds up relative to the liquid. It depends upon fluid properties and flow conditions. There are many “models” for slip (or holdup) in the literature. Hughmark (1962) slip correlation Either horizontal or vertical flow: 1 K 1 x / x ρ / ρ S GL K 1 x / x ρ / ρ GL where 19 K 1 0.12 / Z 0.95 1/ 6 1/ 8 1/ 4 Z NRe N Fr / 1 ε HOMOGENEOUS GAS-LIQUID PIPE FLOW Energy Balance (Eng’q Bernoulli Eqn) 2 f G 2 dx dz m G 2ν ρ g dP ρ DGL dL m dL m dL 2 dν 1 xG G dP where νGL ν G ν L 1 / ρ G 1 / ρ L For “frozen” flow (no phase change): dx 0 dL dν If xG 2 G 1 flow is choked. dP For ideal gas: 1/ k ννGG 1 P1 , 1 k / k P ρP P ρ kP Ts1 For frozen ideal gas/liquid choked flow: ρ kP m Gcρm ε For flashing flow (Clausius-Clapeyron eqn): ν2 cT νG GL p 2 P T λGL Homogeneous Horizontal Flow 2 2fm G dL νGL dx ρ m dz ρD dP m 2 1 xG dνG / dP Flashing Flow - Determine x from (adiabatic) energy balance (or thermo properties database): c T T x p o s λGL Finite Difference Solution – solve for L D2 ΔL - ΔP G2 Δν gΔZ / ν ΣK 2 fit 4f νG m Dimensionless 4f ΔL 2 m ΣK - Δη G*2 Δε gΔZ / P ν ε fit o o *2 D εG where Δη ΔP / Po * GG/Poρ o G ν o / P o ε ν / νoo ρ / ρ L= ΔL i i Using Z Lcos , where θ is the pipe inclination angle with the vertical for horizontal cos = 0 for vertical up flow = 1 for vertical down flow = -1 *2 *2 4f ΔL 2 Δη G Δε / εG ΣK fit m - D 1 gDcosθ / 4 fPoo ν ε Procedure: Find G, Given L, Poe and P . Select desired Δ P and determine at each pressure step from P o to P e . Assume a value for G * . Calculate ΔL at each pressure step. At choke point, ΔL 0 . Adjust until ΣΔL L Ex: Flashing Water in Pipe Given: G,Po ,T o , x o ,s o .ρ Go , ρ Lo Calculate: Pressure Drop over L 1. Assume a value for dL ΔL ; 2. Est. f mm fn DG / μ (Moody, Churchill, ~0.005) 3. ΔP G2 2 f ΔL / ρ D ρ fn ρ , ρ , x,S 1 m m , m L G 11 4. PP ΔP x,T,ρ,ρ Δx,ν 1 o 1 1s1G1L1s 1GL1 ρρ GL11 PPΔP 5. ΔP2 2 1 2 Choked if ΔP Separated Pipe Flows Each Phase Occupies a Specific Fraction of the Flow Area “Two-Phase Multiplier” for friction loss: PP 2 ΦR LL fm fR Reference Single-Phase Flow: G m / A G R = L Total flow is liquid: Lm R = G Total flow is gas: GGm m / A G R = L Gm Total flow is liquid in mixture: GLm 1 x G R = G Gm Total flow is gas in the mixture GGm xG Lockhart-Martinelli (1949) PP 2 ΦLm LL f fLm or: PP 2 ΦGm LL f fGm L-M Two-Phase Multipliers C1 Φ12 Lm χχ2 22 ΦGm 1 Cχ χ State Liquid Gas C tt turbulent turbulent 20 vt laminar turbulent 12 tv turbulent laminar 10 vv laminar laminar 5 L-M Correlating Parameter PP χ 2 LL ffLm Gm 2 2 P 2 fLm 1 x G L fLm ρDL 22 P 2 fGm x G L fGm ρDG Friction Factors f Lm is based on “liquid only” Reynolds No. 1 x GD N ReLm μ L f Gm is based on “gas-only” Reynolds No. xGD N= ReGm mG Duckler et al. (1964) ρ Φ2 L α φ β φ Lm ρ and 2 ρG ΦGm α φ β φ ρ where φ 1 ε , ρ are “no slip” values and lnφ α φ 1 2 1.281 0.478 lnφ 0.444 lnφ 34 0.094 lnφ 0.00843 lnφ 2 ρ φ2 ρ 1 φ βφ L G ρ φ ρ 1 φ mm The Reynolds Number is based on mixture properties: DG N βφ Rem μm Sizing Relief Valves for Two-Phase Flow m A Gvalve GKGvalve d ideal nozzzle Assume Homogeneous Gas-Liquid Mixture in an Isentropic Nozzle 1/ 2 Pn dP GKρ2 dn ρ Po Discharge Coefficient ( K d ) Values given by manufacturer, or in the “Red Book” If flow is choked (critical) use : Kd ,gas If flow is not choked (sub-critical) use : Kd ,liquid TWO-PHASE DENSITY ρ ερGL 1 ε ρ Where ε is the volume fraction of gas: x ε x S 1 x ρGL / ρ x = mass fraction of gas phase (quality). S = slip ratio = vG / vL = (fn(x, ρL/ ρG, …etc) Flashing Flow Non-Equilibrium If L 10 cm Flashing is not complete if L 10 cm In this case, use L x xo x e x o 10 L = nozzle length (cm) x o = initial quality entering nozzle x e = local quality assuming equilibrium If xo > 0.05, x = xe Determine Quality, x e fn( P ) • The quality is determined as a function of pressure by an energy balance on the fluid along the flow path. • The path is usually assumed to be isentropic. HDI – Homogeneous Direct Integration Exact Solution – Based on Numerical Finite Difference Equivalent of Nozzle Equation 1/ 2 1/ 2 Pn dP j n-1 P-P G ρ K -2 ρ K -4 j 1 j n n d n d ρjo ρj 1 ρ j Po Required Information: ρ vs P at constant s from Po to Pn in increments of Pj to Pj+1 . Can be generated from an EOS or from a database (e.g. steam tables). (If choked, Gn Gmax at Pn=Pc) Experimental Data TABLE I VALVE SPECIFICATIONS (Lenzing, et al, 1997, 1998) Valve KdG KdL Orifice Dia. Orifice Area (mm) B&R DN25/40 0.86 0.66 20 0.4869 (Bopp & Reuther Si63) ARI DN25/40 0.81 0.59 22.5 0.6163 (Albert Richter 901/902) 1 x 2 “E” 0.962 0.729 13.5 0.2219 (Crosby JLT/JBS) Leser DN25/40 0.77 0.51 23 0.6440 (441) TABLE II FLOW CONDITIONS (Lenzing, et al, 1997, 1998) Fluid Nom. Pressure Po Pb (bar) (psia) (psia) Air/Water 5 72.495 14.644 Air/Water 8 115.993 14.644 Air/Water 10 144.991 14.644 Steam/Water 5.4 78.295 14.644 Steam/Water 6.8 98.594 14.644 Steam/Water 8 115.993 14.644 Steam/Water 10.6 153.690 14.644 Air-Water (Frozen) Flow .Four Different Valves .Three Different Pressures ARI DN25/40, Air/Water Calc 5 bar Data 5 bar 25000 Calc 8 bar 20000 Data 8 bar ) 2 15000 (kg/s m (kg/s 10000 dG K 5000 0 0.0001 0.001 0.01 0.1 1 xo LESER DN25/40, Air/Water Calc 5 bar Data 5 bar 25000 Calc 8 bar Data 8 bar 20000 Calc 10 bar ) 2 Data 10 bar 15000 (kg/s m 10000 dG K 5000 0 0.0001 0.001 0.01 0.1 1 xo B&R DN25/40, Air/Water Calc 5 bar Data 5 bar Calc 8 bar 35000 Data 8 bar 30000 Calc 10 bar Data 10 bar ) 25000 2 20000 (kg/s m 15000 dG K 10000 5000 0 0.0001 0.001 0.01 0.1 1 xo Crosby 1x2 E, Air/Water Calc 5 bar Data 5 bar 25000 20000 ) 2 15000 (kg/s m 10000 dG K 5000 0 0.0001 0.001 0.01 0.1 1 xo HNDI – Homogeneous Non-Equilibrium Direct Integration For flashing flows, equilibrium is not reached until flow path length reaches 10 cm or more. For L<10 cm, quality (x = gas mass fraction) is lower than it would be at equilibrium (xe). For L < 10 cm, quality is estimated from x=x+ x-x L/10 o e o where xo is the initial (L = 0) quality (L in cm) If xo > 0.05, x = xe Steam-Water Flashing (non-Equilibrium) Flow .One Valve - Leser 25/40 .