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Ref.: Brill & Beggs, Two Phase Flow in Pipes, 6th Edition, 1991. Chapter 3. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) 1- Flow regimes boundaries: The flow regimes map is shown in Figure 3-10. The flow regimes boundaries are defined as a functions of the dimensionless quantities: Ngv, NLv, Nd, NL, L1, L2, Ls and Lm where: - Ngv, NLv, Nd and NL are the same as Hagedorn & Brown method. - Ls= 50 + 36 NLv and Lm= 75 + 84 NLv0.75 - L1 and L2 are functions of Nd as shown in Figure 3-11. Bubble Flow Limits: 0 ≤ Ngv ≤ L1 + L2 NLv Slug Flow Limits: L1 + L2 NLv ≤ Ngv ≤ Ls Transition (Churn) Flow Limits: Ls < Ngv <Lm Annular-Mist Flow Limits: Ngv > Lm Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) 2- Pressure gradient due to elevation change: The procedure for calculating the pressure gradient due to elevation change in each flow regimes is: - Calculate the dimensionless slip velocity (S) based on the appropriate correlation - Calculate vs based on the definition of S: - Calculate HL based on the definition of vs : vs v sg 1 H L v sL HL HL vs S ( L g ) / L 4 v s v m ( v m v s ) 4 v s v sL 2 2vs - Calculate the pressure gradient due to elevation change: g dP s where gc d Z elevation s LH L gH g 0 .5 Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) Correlations for calculating S in each flow regimes: Bubble Flow: S F1 F 2 N Lv N gv ' F3 1 N Lv 2 where F3 F3 ' F4 Nd F1 , F2 , F3 and F4 can be obtained from Figure 3-12. Slug Flow: 0 . 982 S (1 F5 ) N gv F6 ' (1 F7 N Lv ) 2 where F6 0 . 029 N d F6 ' F5 , F6 and F7 can be obtained from Figure 3-14. Mist Flow: Duns and Ros assumed that with the high gas flow rates in the mist flow region the slip velocity was zero (ρs= ρn). Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) 3- Pressure gradient due to friction: Bubble Flow: f tp L v sL v m dP 2 gc d d Z friction f1 is obtained from Moody diagram ( N Re where L v sL d L f tp f 1 f 2 / f 3 ), f2 is a correction for the gas-liquid ratio, and is given in Figure 3-13, and f3 is an additional correction factor for both liquid viscosity and gas-liquid ratio, and can be calculated as: f 3 1 f1 v sg 50 v sL Slug Flow: The same as bubble flow regime. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) Annular-Mist Flow: In this region, the friction term is based on the gas phase only. Thus: f tp g v sg dP 2 gc d d Z friction 2 where d d , v sg v sg d 2 d 2 As the wave height on the pipe walls increase, the actual area through which the gas can flow is decreased, since the diameter open to gas is d – ε. After calculating the gas Reynolds number, N Re g v sg d g , the two- phase friction factor can be obtained from Moody diagram or rough pipe equation: f tp 4 4 log 1 ( 0 . 27 / d ) 2 10 0 . 067 d 1 . 73 for d 0 . 05 Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) Duns and Ros noted that the wall roughness for mist flow is affected by the wall liquid film. Its value is greater than the pipe roughness and less than 0.5, and can be calculated as follows (or Figure 3-15): for N W e N 0 . 005 : for N N 0 . 005 : We d 0 . 0749 L g v sg d 2 0 . 3713 L ( N W e N ) 0 . 302 g v sg d 2 d Where g v sg 2 N we (Weber number ) L L 2 , N L L Duns and Ros suggested that the prediction of friction loss could be refined by using d – ε instead of d. In this case the determination of roughness is iterative. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) 4- Pressure gradient due to acceleration: Bubble Flow: The acceleration term is negligible. Slug Flow: The acceleration term is negligible. Mist Flow: v m v sg n d P dP or g c P d Z total d Z acc dP d Z total dP dP d Z ele d Z f 1 Ek Where Ek v m v sg n gc P Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) Transition Flow: In the transition zone between slug and mist flow, Duns and Ros suggested linear interpolation between the flow regime boundaries, Ls and Lm , to obtain the pressure gradient, as follows: dP dP dP A B d Z Transition d Z Slug d Z Mist Where A L m N gv Lm Ls , B N gv L s Lm Ls 1 A Increased accuracy was claimed if the gas density used in the mist flow pressure gradient calculation was modified to : ' g N gv g Lm Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) Orkiszewski, after testing several correlations, selected the Griffith and Wallis method for bubble flow and the Duns and Ros method for annular-mist flow. For slug flow, he proposed a new correlation. Bubble Flow 1- Limits: vsg / vm < LB Where L B 1 . 071 0 . 2218 v m / d and L B 0 . 13 2 2- Liquid Holdup: HL vm 1 0 . 5 1 vs (1 v m / v s ) 4 v sg 2 / vs Where the vs have a constant value of 0.8 ft/sec. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) 3- Pressure gradient due to friction: f tp L v L dP 2 gcd d Z friction 2 Where ftp is obtained from Moody diagram with liquid Reynolds number: N Re L vL d L 4- Pressure gradient due to acceleration: is negligible in bubble flow regimes. Slug Flow 1- Limits: vsg / vm > LB and Ngv < Ls Where Ls and Ngv are the same as Duns and Ros method. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) 2- Two-phase density: s L ( v sL v b ) g v sg v m vb L The following procedure must be used for calculating vb: 1- Estimate a value for vb. A good guess is vb = 0.5 (g d)0.5 2- Based on the value of vb , calculate the N Re b L vb d L 3- Calculate the new value of vb from the equations shown in the next page, based on NReb and NReL where N Re L L vm d L 4- Compare the values of vb obtained in steps one and three. If they are not sufficiently close, use the values calculated in step three as the next guess and go to step two. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) Use the following equations for calculation of vb: v b 0 . 546 8 . 74 10 v b 0 . 35 8 . 74 10 0 . 251 8 . 74 10 where 6 6 6 N Re L N Re L N Re L g d g d g d N Re b 3000 for N Re b 8000 for for 3000 N Re b 8000 2 13 . 59 L v b 0 . 5 0 .5 d L 0 .5 Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) The value of δ can be calculated from the following equations depending upon the continuous liquid phase and mixture velocity. Continuous Value Liquid Phase of vm Equation of δ Water < 10 Water >10 0 . 013 log( L ) d 0 . 045 log( L ) d Oil <10 0 . 799 >10 0 . 681 0 . 232 log( v m ) 0 . 428 log( d ) 0 . 709 0 . 162 log( v m ) 0 . 888 log( d ) 0 . 0127 log( L 1) d Oil 1 . 38 1 . 415 0 . 284 0 . 167 log( v m ) 0 . 113 log( d ) 0 . 0274 log( L 1) d 1 . 371 0 . 161 0 . 569 log( d ) X 0 . 01 log( L 1) X log( v m ) 0 . 397 0 . 63 log( d ) 1 . 571 d Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) Data from literature indicate that a phase inversion from oil continuous to water continuous occurs at a water cut of approximately 75% in emulsion flow. The value of δ is constrained by the following limits: a ) For v m 10 : 0 . 065 v m b ) For v m s 1 10 : vm vb L vb These constraints are supposed to eliminate pressure discontinuities between equations for δ since the equation pairs do not necessarily meet at vm=10 ft/sec. Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Orkiszewski) 3- Pressure gradient due to friction: f tp L v m v sL v b dP 2 g c d v m vb d Z friction 2 Where ftp is obtained from Moody diagram with mixture Reynolds number: N Re L vm d L 4- Pressure gradient due to acceleration: is negligible in slug flow regime. Transition (Churn) Flow Limits: Ls < Ngv <Lm The same as Duns and Ros method. Annular-Mist Flow Limits: Ngv > Lm The same as Duns and Ros method. Two-Phase Flow Correlations Beggs and Brill Beggs and Brill method can be used for vertical, horizontal and inclined two-phase flow pipelines. 1- Flow Regimes: The flow regime used in this method is a correlating parameter and gives no information about the actual flow regime unless the pipe is horizontal. The flow regime map is shown in Figure 3-16. The flow regimes boundaries are defined as a functions of the following variables: 2 N Fr vm L1 316 L 0 . 302 , gd 1 . 4516 L 3 0 . 10 L , L 2 9 . 252 10 , 6 . 738 L 4 0 .5 L 4 2 . 4684 L Two-Phase Flow Correlations Beggs and Brill Segregated Limits: L 0 . 01 and N Fr L1 or L 0 . 01 and N Fr L 2 Transition Limits: L 0 . 01 and L 2 N Fr L3 Intermittent Limits: 0 . 01 L 0 . 4 and L 3 N Fr L1 or L 0 . 4 and L 3 N Fr L 4 Distributed Limits: L 0 . 4 and N Fr L1 or L 0 . 4 and N Fr L 4 Two-Phase Flow Correlations Beggs and Brill 2- Liquid Holdup: In all flow regimes, except transition, liquid holdup can be calculated from the following equation: a L b H L ( ) H L ( 0 ) , H L (0) N c Fr : H L (0) L with constraint Where HL(0) is the liquid holdup which would exist at the same conditions in a horizontal pipe. The values of parameters, a, b and c are shown for each flow regimes in this Table: Flow Pattern a b c Segregated 0.98 0.4846 0.0868 Intermittent 0.845 0.5351 0.0173 Distributed 1.065 0.5824 0.0609 For transition flow regimes, calculate HL as follows: H L (transitio n) A H L (segregate d) B H L (intermitt , A ent) L 3 N Fr L3 L 2 , B 1 A Two-Phase Flow Correlations Beggs and Brill The holdup correcting factor (ψ), for the effect of pipe inclination is given by: 3 1 C sin( 1 . 8 ) 0 . 333 sin (1 . 8 ) Where φ is the actual angle of the pipe from horizontal. For vertical upward flow, φ = 90o and ψ = 1 + 0.3 C. C is: C (1 L ) ln d L N Lv N Fr , with restrictio n that C 0 . e f g The values of parameters, d’, e, f and g are shown for each flow regimes in this Table: Flow Pattern d' e f g Segregated uphill 0.011 -3.768 3.539 -1.614 Intermittent uphill 2.96 0.305 -0.4473 0.0978 Distributed uphill All patterns downhill No correction 4.70 -0.3692 C=0,ψ=1 0.1244 -0.5056 Two-Phase Flow Correlations Beggs and Brill 3- Pressure gradient due to friction factor: dP dL 2 f tp n v m , 2 gc d f f tp f n e S fn is determined from the smooth pipe curve of the Moody diagram, using the following Reynolds number: N Re n v m d The parameter S can be calculated as follows: n For 1 y L / H L2 ( ) 1 . 2 S ln( 2 . 2 y 1 . 2 ) and for others: S ln y 0 .0523 3 . 182 ln y 0 . 8725 (ln y ) 0 . 01853 (ln y ) 2 4 Two-Phase Flow Correlations Beggs and Brill 4- Pressure gradient due to acceleration: Although the acceleration term is very small except for high velocity flow, it should be included for increased accuracy. s v m v sg d P dP or g c P d L total d L acc dP d L total Where Ek dP dP d L ele d L f 1 Ek v m v sg s gc P , g dP s sin gc dL ele Figure 3-10. Vertical two-phase flow regimes map (Duns & Ros). F3 F4 F2 F4 F5 F6 Figure 3-16. Beggs and Brill, Horizontal flow regimes map.