A THEORETICAL STUDY OF INTERFACIAL WAVES IN CHURN FLOW

Transcription

1 1 37 th European Two-Phase Flow Group Meeting A THEORETICAL STUDY OF INTERFACIAL WAVES IN CHURN FLOW J. R. BARBOSA, Jr. and G. F. HEWITT Department of Chemical Engineering & Chemical Technology, Imperial College of Science, Technology & Medicine, London, SW7 2BY.

2 2 37 th European Two-Phase Flow Group Meeting CONTENTS Objectives and methods; Inception of churn flow; Mechanism of churn flow; Govan et al. (1990) visualisation experiments; Theoretical model; Results; Conclusion.

3 3 37 th European Two-Phase Flow Group Meeting OBJECTIVES AND METHODS To describe the inception of interfacial waves characteristic of churn flow (Hewitt et al., 1985); To predict the experimental data obtained by Govan et al. (1990) for wave growth in a specially mounted test section; To use a simple model for wave growth based on macroscopic mass and momentum balances.

4 4 37 th European Two-Phase Flow Group Meeting INCEPTION OF CHURN FLOW Entrance effect, unstable plug flow (Taitel et al., 1980); Excessive aeration of the liquid slug (Mishima and Ishii, 1984; Brauner and Barnea, 1986); Flooding in Taylor bubble (Nicklin and Davidson, 1962; McQuillan and Whalley, 1985; Jayanti and Hewitt, 1992)

5 5 37 th European Two-Phase Flow Group Meeting MECHANISM OF CHURN FLOW Wave growth near the injector and levitation; Flow reversal (falling film) between waves; Next wave picks up the falling film and re-accelerates it. Hewitt et al. (1985)

6 6 37 th European Two-Phase Flow Group Meeting GOVAN ET AL. (1990) EXPERIMENTS Direct evidence of wave formation mechanism; Specially constructed transparent liquid inlet; High-speed video recording of wave formation (wave frequency, velocity and distance travelled).

7 7 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - i Macroscopic balances of mass and momentum; Coherent wave growing at the bottom of the porous section at a rate pipe wall porous sinter! M LF, in falling film CV M! M! LF, in LF, out U w! M LF, out r z

8 8 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - ii Conservation equations d dt V M! M! ρ = L w LF, in LF, out d F = ρ ( ) dt U V U M!,, + U, M!, z L w w LF in LF in LF out LF out Unknown variables F, V, M!, M!, U. z w LF, in LF, out w

9 9 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - iii Wave shape and volumetric relationships δ s, in δ w ( z) A w L w δ w ( ) z δ = δ w1 ; 0 z L 2 ( L z) 1 δ ( δ ) ( δ ) 2π w 2 w1 = Aw + s, out + Aw s, out cos 2 L w 1 2π δw2 = ( Aw + δs, in ) ( Aw δs, in ) cos 2 w ; L 2 < z L w1 w w ( L z) L w w 0 z δ s, out L w V d ( z) w = π T δw dz d dt V = 1 d L d 2 π dt A 0 w T w w

10 10 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - iv Closure relationship for film thickness M! LF, Nu ( ) 3 T L L G s π d gρ ρ ρ δ = 3η L ( ) M! = π d ρ δ U + U LF, in T L s, in LF, in w ( ) M! = π d ρ δ U + U δ LF, out T L s, out LF, out w 4 s w s LF Nu + BU δ C M!, = 0 (Nusselt) Linear variation with distance for M!! LF in., and M LF, out

11 11 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - v Force balance over the wave (Shearer and Davidson, 1965) Gravitational force M! + M! LE G Pressure force on the core δ s, in Pressure force on the liquid F gw F sw Wall shear force F pc F pl Interfacial shear force δ s, out Fz = FpC + FpL Fsw Fgw M! + M! LE G U w

12 12 37 th European Two-Phase Flow Group Meeting THEORETICAL MODEL - vi Solution procedure d dt A M! LF in M! w = 2 π d ρ L d dt U d dt z w = = U w 4 F, LF, out z T L w M! 2 LF out M! 2, LF, in + π d ρ δ π d ρ δ T L s, out ( 2 +, +, ) π d ρ L A δ δ T L s, in T L w w s in s out ( ) ( ) A ( 0) = A ; U 0 = 0; z 0 = 0. w w, crit w Make: Lw, e, M! LF, out vary over a wide range of values, solving the differential equations at each iteration; Set: M!! LF, in = ML( 1 e) Best solution: smallest deviation to data; Costigan (1997) analysis gives A w, crit.

13 13 37 th European Two-Phase Flow Group Meeting RESULTS - i D i s t a n c e [ m ] ( ) 10 3 kg / s Sequence 1 M!. ;! G = 8 3 ML = 27 Sequence 2 M!. ;! G = 8 3 ML = 42 Sequence 3 M!. ;! G = 6 8 ML = 103 Sequence 4 M!. ;! G = 4 5 ML = 105 Sequence 5 M! = 3. 6; M! = 108 G L S 2 S 3 S 1 S 4 S 5 Effect of gas velocity; Lower gas flowrates give longer growth and acceleration periods T i m e [ s ]

14 14 37 th European Two-Phase Flow Group Meeting RESULTS - ii

15 37 th European Two-Phase Flow Group Meeting CONCLUSION Presentation of a simplified macroscopic model for inception of waves characteristic of churn flow; Comparison with the experimental data obtained by Govan et al. (1990) for wave growth in a specially mounted test section; Encouraging agreement for distance travelled by the waves and other wave parameters; Phenomenological prediction of pressure drop is still a long way to go.