Article pubs.acs.org/IECR
Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX-XXX
On Transformation of a Taylor Bubble to an Asymmetric Sectorial Wrap in an Annuli Lokesh Rohilla and Arup Kumar Das* Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, 247667 Roorkee, India ABSTRACT: Genesis of an asymmetric sectorial wrap of air from a Taylor bubble upon facing concentric annular obstruction in a stagnant kerosene column is investigated. Careful observations are reported from processed images taken using high speed camera and with finite volume based simulations. The interfacial reconstruction of a Taylor bubble is completed through six distinct stages, namely, plateau formation, doughnut shape bypass of obstruction followed by nucleation, preferential rise, retraction of the lagging lobe and subsequent thread formation, consumption of thread including bubble segment stage, and finally the manifestation of the annular bubble before rising as steady annular sectorial wrap. Analysis of experimental observations related to interfacial rise and kinematic estimation from numerical study is presented in support of claims related to different transformation stages. A close similarity has also been reported between the stages involved in the interfacial reconstruction process and classical Rayleigh−Taylor instability.
1. INTRODUCTION Slug flow is frequently encountered over a wide range of engineering applications involving liquid and gas interaction inside a vertical pipe. Gaseous slug is characterized by train of bullet shaped bubbles having spherical cap and cylindrical body.1 Slug bubbles formed inside a tube was first identified by Dumitrescu,2 and these are alternatively termed as Taylor bubble due to substantiation by Davies and Taylor.3 Isolated Taylor bubble during a majority of the times (zone V as described in White and Beardmore4) possesses a stable shape and steady up rise of velocity which is solely dependent upon conduit diameter. Train of such bubbles inside a conduit finds numerous applications in cryogenics, energy systems, biological assays, and chemical reactors.5−8 Due to its widespread use in engineering processes, hydrodynamics of the Taylor bubble has been a fertile area of research during the past years. These efforts took experimental routes9−11and established analytical or numerical understanding12−14 of its dynamics. Motion of symmetric Taylor bubble (Nf = (gD3)1/2/ν < 500)15 in a pipe becomes critical when it encounters obstruction in flow path. Such insertions in pipe carrying gas−liquid slug flow are very common in industries like crude oil exploration, double pipe heat exchanger, nuclear reactors with catalyst rods and internal loop air lift chemical reactors.16 In the case of coaxial obstruction in slug flow, one may observe transformation of symmetric Taylor bubble to an open sectorial wrap as around the obstruction by the interface. Azimuthal asymmetric nature of sectorial wrap was first identified by Griffith17 and later thoroughly explained by Kelessidisis and Dukler.18 Their analysis indicated that possible axisymmetric configuration after bypassing obstruction rises at the lower rate than the open sectorial bubble, obtained experimentally. Hence, symmetric © XXXX American Chemical Society
annular bubble around an obstruction is almost never practically observed during the experiments. On the other hand, the asymmetric sectorial bubble wrap takes the ellipsoidal shape at the bubble cap unlike the spherical one in fully developed Taylor bubble. Subsequently, Das et al.16 did combine theoretical and experimental analyses on a sectorial wrap bubble rising in an annuli to propose rise velocity from fundamental physics. Their theoretical model assumed potential flow near the bubble nose region and predicted the velocity as 0.323 g (D1 + D2) , where D1 and D2 denote the corresponding outer and inner concentric tubes diameter, respectively, consisting of the annular shape. In the same year, Hills and Chety19 have also performed experimental observations on sectorial bubble wrap to support findings of Das et al.16 Continuing in their effort, Das et al.20 experimentally showed that the bubble nose and its angle of wrap are independent of the volume. Increase in volume only lengthens the sectorial warp keeping the elliptical nose shape intact.16 Agrawal et al.21 observed the limit of wrapping characteristic of the annular bubble by reducing the inner tube dimension and keeping the outer one same. Their experimental observation showed the merging of the two shoulders of the sectorial bubble diminishing the extent of liquid bridge, separating them. The angle of wrap, calculated for these configurations, was reported to be even more than 360° for prediction of merging nature. They have also experimented asymmetry of bubble Received: Revised: Accepted: Published: A
September 3, 2017 November 9, 2017 November 14, 2017 November 14, 2017 DOI: 10.1021/acs.iecr.7b03663 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
based on auxiliary velocity and updated pressure field. Modified velocity field at forward time step is checked for inconsistency in continuity equation. Piecewise linear interface capturing scheme25 is used for reconstruction of the interface at each time step using data of volume fraction obtained satisfying advection. One may refer to Fuster et al.26 for detailed understanding of the scheme. Open source Gerris solver, developed and maintained by Popinet,27 has been used for simulations. Geris uses multilevel, adaptive tree structure for discretization of Cartesian domain truncated by cylindrical solids. Gerris uses height function method24 for handling large singularity in interface prediction. The approach has potential equivalent to coupled level set-volume of fluid method (CLSVOF) as described by researchers.28−30 Gerris also uses polynomial fitting for evolution of error free smooth interface in extreme cases like CLSVOF. Time step for simulation has been chosen 3 μΔx ⎞ ⎛ as minimum than max⎜ ρΔx , σ ⎟ to keep the Courant πσ ⎝ ⎠ number lesser than 0.5. A tolerance of convergence criterion is set as 10−6 for all the variables, which needs to be achieved within a maximum of 2000 iterations. Outer extent of computation domain is bounded by a cylinder of diameter (Φ) D1 having length L1. For creation of annular space, another cylindrical solid is coaxially places from the top of the computational domain. Inserted cylinder is taken of diameter (Φ) D2 and length L2 (L2 < L1). Lower portion (L1 − L2) of the domain is cylindrical, whereas an annular space ((D1 − D2)/2) between the solids constitutes the upper half. Physical faces of the solids are implied to have no slip and no penetration boundary conditions. Top and bottom boundaries are also taken as wall, and the domain length (L1) is kept large enough so that extreme walls do not have any influence in the reconstruction of interface. Atmospheric pressure outflow at the top and no-flow inlet boundary condition may have resulted in ∼1% change11 in up rise velocity of the bubble. An equilibrium contact angle of 90° has been considered between solid faces of the inset and working fluid. Similar treatment has been also followed for tube outer wall. Figure 1 shows schematic diagram of the overall and working domain along with imposed boundary conditions. Capturing the interface reconstruction depends on proper resolution of mesh near the solids and phase boundary.
inside configuration with square outer and inner tubes. Their results showed that model proposed by Das et al.16 has limitation in predicting bubble velocity for the situations having asymmetry in the annular gap of the conduit around the bubble. It can be observed from literature that symmetric Taylor bubble and sectorial annular wrap have completely different interfacial distribution. Transition between these two configurations and reasons behind interfacial reconstruction are still not clear from fluidic physics. The transformation of a fully developed Taylor bubble to an annular bubble is the amalgamation of the hydrodynamics of both Taylor bubble and annular bubble sector, which makes it rich in physics and important from a practical point of view. Interaction of the fully developed Taylor bubble with the circular bottom face of the inner cylindrical tube initially shows the same behavior as collision of a small air bubble beneath a flat plate.22,23 But establishing analogy between these two subjects requires further effort. In this paper, we have presented combined numerical and experimental efforts for exploring the transformation mechanisms and interface reconstruction in a stagnant kerosene column. Similarity of the reconstruction mechanisms has also been highlighted with gravity driven Rayleigh Taylor instability.
2. MATERIALS AND METHODS 2.1. VOF Based Numerical Simulation. Simulations are performed in 3D framework with finite volume discretization of the domain. Incompressible air and kerosene are taken as working fluids. Summing up body and surface forces in the domain, mass and momentum conservation equations can be written as (1) ∇·u = 0 ρ[∂tu + u ·∇u] = −∇p + ∇·(2μD) + σkδ I n + Fb
(2)
Here, u and p are the velocity vector having three mutual orthogonal components ui and pressure field, respectively. The morphological properties like density (ρ) and viscosity (μ) represent variable density and viscosity field which can be evaluated based on eq 3 as Ψ(c) = Ψ1c + Ψ2(1 − c)
(3)
Occupancy of a particular fluid in a cell is defined by its volume fraction c which takes either 0 (air) or 1 (water) in bulk and inbetween in the interfacial cells. D is the deformation tensor ∂uj ⎞ 1 ⎛ ∂u which is related to the velocity field as Dij = 2 ⎜ ∂xi + ∂x ⎟. ⎝ j i⎠ Surface tension (σ) has been incorporated into the momentum equation as source with interface Kronecker delta (δI), surface curvature (k), and normal vector (n). Fb is the body force affecting the velocity field which in the present effort replicates gravitational field. Volume fraction (c) is advected by using the volume of fluid (VOF) framework as ∂tc + u ·∇c = 0
(4)
Pressure and volume fraction terms are evaluated using second order accurate staggered in time discretization which is coupled with time-splitting projection method.24 Viscous term is discretized using second order accurate Crank−Nicholson scheme, and advection issues are resolved following second order upwind Bell−Colella−Glazs scheme.25The auxiliary velocity at intermediate time is used to update pressure field. Finally, temporal advancement of velocity field is performed
Figure 1. Computation domain for Taylor bubble simulation. B
DOI: 10.1021/acs.iecr.7b03663 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research A thorough analysis of suitability in mesh size has been performed, and steady rise nose velocities of a Taylor bubble under buoyancy pull inside a 20 mm diameter vertical tube, as obtained from simulations, have been reported for four different mesh sizes in Figure 2a. With smallest size mesh of
Table 1. Comparison of Taylor Bubble Velocity by Experiments and Numerical Simulation method
velocity of Taylor bubble (m/s)
experimental numerical
0.1504 0.1513
Taylor bubble obtained by experiments and numerical simulation. After obtaining confidence in prediction tool, simulations are performed for studying stages of interface reconstruction while passing around cylindrical insert. Numerical findings are confirmed from careful experimental investigation, for which an in-house facility was developed. 2.2. Experimental Apparatus. Figure 3 shows the schematic of in-house facility for Taylor bubble visualization.
Figure 2. (a) Grid independence test for the finite volume based discretization scheme for bubble velocity at different refinement levels and inset showing the comparison of experimental shape of the Taylor bubble at six refinement levels. (b) Adaptive mesh refinement in VOF simulations near the interface and walls.
Figure 3. (a) Schematic of the experimental apparatus: (1) frame, (2) Plexiglas tube, (3) view box, (4) concentric insert, (5) Taylor bubble, (6) quick closing valve, (7) fraduated bubble generating section, (8) needle valve, and (9) fluid collection tray. (b) View box for visualization of interfacial reconstruction.
3.125 × 10−4 m, velocity of numerical bubble (0.1513 m/s) reaches within 99.4% accuracy as compared to experiment (0.1504 m/s). The match of the shapes in interface with smallest mesh size of 3.125 × 10−4 m and experimental snapshot is also shown in the inset. Further decrease in mesh size to 1.5625 × 10−4 m only increases accuracy in nose velocity prediction marginally (
