Interfacial Wave Transitions in Liquid-Liquid Flows and Insight into Flow Regime Transition

Transcription

1 Interfacial Wave Transitions in Liquid-Liquid Flows and Insight into Flow Regime Transition M. J. McCready, B. D. Woods & M. R. King Chemical Engineering University of Notre Dame /

2 Flow geometry of interest Two-fluid stratified flow gas liquid Transition from a stratified state to large disturbances occurs by linear and nonlinear processes. This talk will discuss these mechanisms and applications to other systems

3 Motivation of this research At present, predictive capabilities for multiphase flows are no where near as good as for single phase flows Single phase: 2 dimensionless groups, Two-phase 6 dimensionless groups We would like to improve understanding of nonlinear processes in stratified flow geometries to enable Better prediction of flow regimes for process and pipeline flows. Precision chemical reaction processing that is possible only with precise control of contacting times.

4 Specific Applications Slug formation in two-phase pipe flow Slugs form from waves through combined linear and nonlinear processes Formation of pulses in packed beds Krieg, Helwick, Dillon and McCready (AIChE J, 1995) have shown significant similarity between pulses and slugs Not yet been exploited in modeling/prediction Coherent disturbance generation in arrays of microreactors or structured packing. Larger transport coefficients should be available if sufficient interconnectivity is cleverly done.

5 Flow regime importance Why do we care which regime it is? Pressure drop varies by an order of magnitude for same flow rates but different geometric configurations, Factor of >20 for idealized laminar flow, /Effect.of.Flow.Regime.nb Atomization can be be 40% of total liquid, Big effect on wall heat transfer Slugs cause large flow surges and pressure variations

6 Status of predictive methods for flow regimes Two methods are employed, Stability of an existing slug Linear stability of the stratified flow to long waves The first procedure has been shown to work reasonably well for thin liquid layers. The second is a necessary, but not sufficient condition and has not been calculated correctly in the literature Important nonlinear processes occur and are not accounted for in standard predictive methods

7 Formation of a slug from waves < Flow direction a b c d e f g h i j Data of Woods, 1998

8 Slug formation from waves Two period doublings occur U SG = 1.8 m/s U SL = 0.18 m/s h/d =.6 Data of Woods, 1998

9 Some flow regime transition models

10 Specific Applications Slug formation in two-phase pipe flow Slugs form from waves through combined linear and nonlinear processes Formation of pulses in packed beds Krieg, Helwick, Dillon and McCready (AIChE J, 1995) have shown significant similarity between pulses and slugs Not yet been exploited in modeling/prediction Coherent disturbance generation in arrays of microreactors or structured packing. Larger transport coefficients should be available if sufficient interconnectivity is cleverly done.

11 Gas-Liquid Flow in a packed bed with Pulses Gas Liquid Liquid partial pulses Full pulses Wu, McCready, Varma, 1995, 1998, show substantial effect of pulses on reaction products

12 Pressure tracings of pulses L = 7.9 kg/m 2 - s air-water 10 cm column 1m from inlet 0.3 cm spheres

13 Array of gas-liquid microreactors Extra connections could allow formation of strong disturbances that would not occur in individual small channels Connections beween channels Gas Liquid Gas Liquid Gas Liquid Gas Liquid

14 Issues to be resolved in this talk Given the presence of nonlinear processes, is linear stability of long waves a useful criteria for flow regime transition It is commonly used now Which theoretical procedure can help explain and predict the nonlinear processes that are occurring?

15 Experiments Need these to be laminar flow Effectively means that they need to be liquid-liquid flows Need to be able to handle the issue of development with distance. Two systems were used Matched density, rotating Couette flow Oil-water channel flow

16 Rotating, Two-(matched density) liquid, Couette Flow, Wave Experiment Outside cylinder is Plexiglas, Inside cylinder is Aluminum painted black Torque transducer Hg Layer Outside cylinder is rotated.

17 Two-(matched density) liquid, rotating Couette device laser mirror camera mercury Couette Cell

18 Rotating Couette experiment

19 Theoretical analysis (linear laminar theory) Yih, 1967, Blennerhassett, 1980, Yiantsios & Higgins, 1988 The complete differential linear problem can be formulated as U = F'(y) Exp[i k (x-c t)], u = f'(y) Exp[i k (x-c t)], V = - i k F (y) Exp[i k (x-c t)], v = - i k f(y) Exp[i k (x-c t)] where F(y) and f(y) are the disturbance stream functions F = F' = [1a] f = [1b] f' - u b f = F' - Ub [1c] ub ( 0) - c u b ( 0) - c f'' + k 2 f = m (F'' + k 2 [1d] 1 nr (f''' - 3 i k f k2 f') + i k (f u b ' - f' (u b (0) - c)) + (F + k2 T) (u b (0) - c) R 2 = r R (F''' - 3 i r k f k2 F') + r i k (F U b ' - F' s) + F (u b (0) - c) R [1e]

20 Theoretical analysis (linear laminar theory) i k (U b -c) (F'' - k 2 F) - i k U b '' F = R -1 (F iv - 2 k 2 F''+ k 4 F), for 0 y 1 [1f] i k (u b -c) (f'' - k 2 f) - i k u b '' f = (n R) -1 (f iv - 2 k 2 f''+ k 4 f), for -1/d y 0 [1g] f = f' = y = -d -1. [1h] viscosity ratio ==> m = m 2 /m 1, density ratio ==>r = r 2 /r 1, ratio of kinematic viscosities ==> n, s = u b (0) -c depth ratio ==> d = D 2 /D. 1 liquid average velocity profiles ==> u b wavenumber ==> k gas velocity ==> U b

21 Types of linear growth curves Growth rate Growth rate Wavenumber Wavenumber Growth rate Wavenumber

22 Wave map for rotating Couette flow experiment Gallagher, Leighton and McCready, Phys. Fluids,1997 Left of the line, only short waves are unstable Regions of "no waves" exist where long waves are unstable plate speed (cm/s) stable long waves steady 2-D waves occur in most of this range Growth rate No waves steady periodic unsteady waves solitary long wave stability boundary 0.2 avenumber Growth rate Atomization h Wavenumber 0.8

23 Weakly-nonlinear theory Du Dt = P + 2 u Spectral reduction of Navier-Stokes equations and boundary conditions Φ' ( xyt,, ) = f( yt, )expiα x n n= Φ' ( xyt,, ) = A ( t) φ ( xy, ) = A ( t) φ ( y)expiα x nl nl n= l= 1 n= l= 1 nl n nl n A nl Complex Amplitude Function φ nl Linear eigenfunctions

24 Weakly-nonlinear theory Center manifold projection to produce a Stuart - Landau equation. Renardy & Renardy, (Blennerhassett, 1980 also obtained this by multiscale analysis) A = LkA t + 2 ( ) β A A A= complex wave amplitude β= Landau Constant L(k) = linear eigenvalue

25 Experimental verification rotating Couette experiment wave amplitude 1 x10 3 [m] increasing speed decreasing speed theory for maximum growing wavelength theory for observed wavelength U [m/s] 0.30 viscosity ratio = 50 density ratio = depth ratio = M. Sangalli, C. T. Gallagher, D. T. Leighton, H. -C. Chang and M. J. McCready Phys. Rev. Lett. 75, pp (1995).

26 Weakly- nonlinear simulations A series of amplitude coupled amplitude equations obtained from the spectral expansion is integrated. nl nl nl nl, pr, qs pr pqrs,,, A = L A + ϑ A A + ξ A A A qs pqmrsz,,,,, nl, pr, qs, mz pr qs mz Both the dynamic and steady state behavior are watched.

27 Wave map for rotating Couette flow experiment Regions of "no waves" exist where long waves are unstable 50 Atomization plate speed (cm/s) stable long waves steady 2-D waves occur in most of this range No waves steady periodic unsteady waves solitary long wave stability boundary h Will show spectral simulation at this condition

28 Couette flow linear growth rate growth rate TextEnd Depth ratio=1.5 Viscosity ratio = 55 Density ratio=1 Rotation rate=15 cm/s scaled wavenumber

29 Spectral simulation, Couette flow A m p l i t u d e Wavenumber King and McCready, Phys. Fluids, 2000

30 Couette flow simulations Evolution of wave spectrum with time. No preferred wavenumber exists. Should explain why we see no waves amplitude t=300 t=243 t=183 t=122 t= wavenumber

31 Experiments laminar, oil-water channel flow 2.44 m.5-1 cm Oil phase Water phase } Tracings PSD CSD - Wave speeds Bicoherence.2 mm platinum wire probes 30 khz Signal 1 cm

32 5 4 3 Measured wave transitions Linear growth: interfacial mode "shear" mode wave spectrum ω (1/s) 2 1 Re oil = 3 Re water = 650 φ xx (cm 2 /s) f (Hz)

33 Measured wave transitions ω (1/s) (linear growth) Linear growth: interfacial mode "shear" mode measured spectrum Re oil =3 Re water = φ xx (cm 2 /s) (wave spectrum) f (Hz)

34 Channel flow experiment H = 1cm W = 16 cm Length = 240 cm 10 mw HeNe Laser Paraffinic Mineral Oil µ = 18 cp ρ = 855 g/cm3 Planoconvex Lens Neutral Density Filter Water 100 mm Biconvex Lens Position Sensing Detector

35 Developing oil-water channel flow

36 Simulation of oil-water channel flow A m p l i t u d e Weakly-nonlinear spectral simulation B. D. Woods, M. J. McCready Oil-water channel flow viscosity ratio = 18 density ratio =.88 Reoil=5.5 Rewater = 900 Wavenumber

37 Bicoherence spectrum at 60 cm bicoherence at 60 cm f 2 f2 TextEnd f1 f 1 Bicoherence uses raw FFT data to verify coherence between the phase of different wave frequencies. Such coherence is expected only when nonlinear interactions are occurring.

38 Comparison of oil-water experiments and simulation 10-2 Linear stability Re W =1200 Re W =1200 Re W =650 Re W = wave amplitude linear growth rate (1/s) Simulated Spectra Re W =650 Re W = wavenumber (1/m) -4

39 Growth of wave spectrum with distance wave spectrum (cm 2 -s) Measurement location 20 cm 35 cm 50 cm 100 cm frequency (Hz)

40 Simulated spectra 10-1 Re oil =3.5 Re water = amplitude spectrum Wave amplitude simulation t=20 t=40 t=60 t=80 t=100 t=120 t=140 t=160 t=180 t=200 Linear growth rate grd temporal growth (1/s) frequency (Hz)

41 Quadratic coefficients from simulation

42 Conclusions All evidence is that long wave instability is a necessary ( engineering ) condition for formation of slugs and roll waves It is not a sufficient condition Couette experiment shows conditions where this criterion produces no visible waves Traditional (linear) calculation procedures are not consistent with the physics or with each other

43 Conclusions (cont.) The transition to large waves and slugs shows evidence of period doubling and longwave-short wave interactions For laminar flows, weakly-nonlinear mode equations, solved by integration, seem to describe the qualitative processes Examination of the coefficient spectrum (King and McCready, Phys. Fluids, 2000) provides insight May be able to make progress on turbulent flows because the coefficients are not too sensitive to velocity profile shape