#### Transcript Two Phase II

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: vs S 4 ( L g ) / L - Calculate HL based on the definition of vs : vsL vs vm (vm vs ) 2 4vs vsL vs HL 1 H L H L 2vs vsg - Calculate the pressure gradient due to elevation change: g dP s where s L H L g H g dZ elevation g c 0.5 Two-Phase Flow Correlations Vertical Upward Flow Pipeline (Duns & Ros) Correlations for calculating S in each flow regimes: Bubble Flow: 2 N gv F ' where F3' F3 4 S F1 F2 N Lv F3 Nd 1 N Lv F1 , F2 , F3 and F4 can be obtained from Figure 3-12. Slug Flow: S (1 F5 ) 0.982 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 vsL vm dP 2 gc d dZ friction where f tp f1 f 2 / f 3 f1 is obtained from Moody diagram ( N Re L vsL d ), f2 is a correction L 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 vsg 50 vsL 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 vsg2 d2 dP where d d , vsg vsg 2 2 gc d d dZ friction 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 vsg d , the twog phase friction factor can be obtained from Moody diagram or rough pipe equation: 1.73 1 f tp 4 0.067 2 d 4 log 10 (0.27 / d ) 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 NW eN 0.005 : for N N 0.005 : We Where d d 0.0749 L g vsg2 d 0.3713 L ( NW eN ) 0.302 g vsg2 d g vsg2 L2 N we (Weber number ) , N L 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: vm vsg n dP dP or g c P dZ total dZ acc dP dP dZ ele dZ f dP 1 Ek dZ total Where Ek vm vsg 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 dZ Transition dZ Slug dZ Mist Where A Lm N gv Lm Ls , B N gv Ls 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 vm2 / d and LB 0.13 2- Liquid Holdup: vm 2 H L 1 0.5 1 (1 vm / vs ) 4vsg / vs 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: 2 f v d P tp L L 2 gc d dZ friction 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 (vsL vb ) g vsg L vm vb 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 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 vm d L 4- Compare the values of vb obtained in steps one and three. If they 2- Based on the value of vb , calculate the N Re b L 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: vb 0.546 8.74 106 N Re L vb 0.35 8.74 106 N Re L 0.251 8.74 106 N Re L gd gd gd for N Reb 3000 for N Reb 8000 for 3000 N Reb 8000 0.5 13.59 L 2 where vb 0.5 0.5 L d 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 Oil <10 0.013 log( L ) 0.681 0.232 log( vm ) 0.428 log( d ) d 1.38 0.045 log( L ) 0.709 0.162 log( vm ) 0.888 log( d ) d 0.799 0.0127 log( L 1) 0.284 0.167 log( vm ) 0.113 log( d ) 1.415 d 0.0274 log( L 1) 0.161 0.569 log( d ) X d 1.371 0.01 log( L 1) X log( vm ) 0 . 397 0 . 63 log( d ) d 1.571 Oil >10 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 vm 10 : 0.065 vm b) For vm 10 : vb s 1 vm vb L 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 vm2 vsL vb dP 2 g c d vm vb dZ friction 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: N Fr vm2 , L1 316 L0.302 , L2 9.252 104 L2.4684 gd L3 0.10 L1.4516 , L4 0.5 L6.738 Two-Phase Flow Correlations Beggs and Brill Segregated Limits: L 0.01 and N Fr L1 or L 0.01 and N Fr L2 Transition Limits: L 0.01 and L2 N Fr L3 Intermittent Limits: 0.01 L 0.4 and L3 N Fr L1 or L 0.4 and L3 N Fr L4 Distributed Limits: L 0.4 and N Fr L1 or L 0.4 and N Fr L4 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 bL H L ( ) H L ( 0) , H L ( 0) c with constraint : H L ( 0) L N Fr 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 (transition) A H L (segregated) B H L (intermittent) , A L3 N Fr , B 1 A L3 L2 Two-Phase Flow Correlations Beggs and Brill The holdup correcting factor (ψ), for the effect of pipe inclination is given by: 1 C sin( 1.8 ) 0.333 sin 3 (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: g C (1 L ) ln d eL N Lvf N Fr , with restrictio n that C 0. 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: 2 dP f v tp n m dL f , f tp f n e S 2 gc d fn is determined from the smooth pipe curve of the Moody diagram, using the following Reynolds number: N Re n vm d n The parameter S can be calculated as follows: 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) 2 0.01853 (ln y ) 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 vm vsg dP gc P dL acc dP or dL total dP dP dL ele dL f dP 1 Ek dL total Where Ek vm vsg s gc P g dP , s sin dL ele g c 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.