Numerical Study of Air Entrainment and Liquid Film Wrapping around

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Numerical Study of Air Entrainment and Liquid Film Wrapping around a Rotating Cylinder Basanta Kumar Rana,† Arup Kumar Das,*,‡ and Prasanta Kumar Das† †

Department of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India Department of Mechanical and Industrial Engineering, IIT Roorkee, Roorkee 247667, India

‡

ABSTRACT: Entrainment of a gaseous cusp and wrapping dynamics of liquid film around the horizontal rotating roller have been studied in a stratified layer. Finite volume based simulations have been used for numerical prediction of the interface which has been tracked using volume of fluid (VOF) algorithm. After establishing the transient dynamics of the wrapping at receding front and entrainment at the advancing front, we showed the effect of liquid properties on interfacial dynamics. Mutual interplay between centrifugal force due to rotation of the cylinder and viscous damping generates different patterns of entrainment such as cusp formation, entrained air wrapping, ejection of bubble, and penetration of wrap inside entrainment. Variation of wrapped liquid thickness at the receding end is also studied for a wide range of Morton number and rotational speed. Finally, using scale analysis, we showed that viscosity plays a major role in deciding the azimuthal wrap thickness.

1. INTRODUCTION Entrainment of one fluid into other finds wide applications in cases where increase of interfacial area causes enhancement of heat or mass transfer. Engineering examples like jet pump, ejector, aerator, chemical reactors, etc. are developed based on entrainment principle or favor increase of interfacial area for performance enhancement. On the other hand, this phenomenon is undesirable in typical industrial operations such as the pouring of molten glass or metal which often causes degradation in strength due to entrapment of gas. Entrainment of gas is mainly caused by liquid jet impingement in a pool and widely studied by several researchers.1−8 Entrapment of gas into liquid in stratified configuration can also be caused by rotating a solid roller at the interfacial plane. This can be named as rotary entrainment by mimicking the type of prime mover behind the phenomenon. Apart from applications related to increase in interfacial area, rotary entrainment finds another major application in liquid coating to the solid surfaces for aesthetic decoration or protection from weathering. When a solid roller rotates at immiscible stratified layers of liquids, the interface follows the rotational dynamics and wraps a film around the roller leaving it coated. Experimental and analytical investigations are made by researchers in this direction mainly to understand the coating dynamics around the roller. Spiers et al.9 have predicted the thickness of liquid film on a vertically moving plate which is partially submerged in the liquid. They have found three important regions, namely, static meniscus, dynamic meniscus, and constant-film-thickness regions. Tharmalingam and Wilkinson10 have predicted theoretically the wrapping film thickness over the rotating cylinder with different fluid pairs and have © 2016 American Chemical Society

compared with experimental results by considering different immersion and inspection angles. Bolton and Middleman11 have studied the phenomenon and found the critical speed of the roller by which air is entrained through the side surface of roller experimentally. Campanella and Cerro12 have investigated the flow of viscous fluid around the horizontal partially submerged roller. They have studied the effects of immersion angle, roller radius, and roller speed on the wrapping liquid film thickness by using the rapid-flow approximation. Joseph et al.13 have studied the formation of two-dimensional cusped interfaces in both Newtonian and non-Newtonian fluids at low Reynolds numbers. Gradual cusps are observed in Newtonian fluid, whereas sudden crater formation has been reported around the rotating cylinder which is half submerged in non-Newtonian liquid. They have also presented asymptotic analysis of the cusp tip for zero surface tension fluid pairs. Using counter-rotation between two horizontal cylinders, Jeong and Moffatt14 formed free surface cusp between stratified layers and proposed analytic solutions of the radius of curvature by considering cylinders as vortex dipole. In their analysis, they have included surface tension but neglected gravitational field. Continuing these efforts, Lorenceau et al.15 have proposed the functional form of cusp width and radius of curvature as parameter of roller velocity (capillary number, Ca) from their experimental observations. Film flow around a fast rotating cylinder across air−ink interface is studied by Yu et al.16 using volume of Received: Revised: Accepted: Published: 11950

September 8, 2016 October 25, 2016 October 28, 2016 October 28, 2016 DOI: 10.1021/acs.iecr.6b03477 Ind. Eng. Chem. Res. 2016, 55, 11950−11960

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Industrial & Engineering Chemistry Research fluid based numerical simulation. They have explained the effect of viscosity, surface tension, roller speed, and immersion angle of roller on the liquid film and have given one correlation to predict the film thickness applicable for a range of viscosity and angular velocity. None of these studies stressed entrapment inside another fluid which can be a major area of interest for enhancement of heat and mass transfer. Our study reports the numerical effort to find out about film wrapping and liquid entrainment by the rotation of horizontal roller which is placed between gas−liquid interface. First, we have characterized the cusp growth to observe different interfacial structures during angular gas film wrapping. Next, a comprehensive study has been presented to report cusp development and entrainment pattern at different rotational speeds. The effect of different thermophysical properties on wrapping and entrainment is also a topic of this paper. Finally, by use of analytical formulation, wrap thickness has been predicted at different azimuthal plane.

number gives the relative effect between viscous and surface tension forces that are acting on the gas−liquid interface and Morton number mainly represents the viscous resistance to surface tension driven force for obtaining spherical interfacial configuration. 2.2. Formulation of the Problem. Numerical probing of entrainment dynamics has been performed by solving volume of fluid based incompressible Navier−Stokes solver. In this framework, continuity and momentum equations including body and surface forces can be written as

(2)

(3)

Here μ = (μi,μj) is the fluid velocity, ρ ≡ ρ(x,t) is the fluid density, and μ ≡ μ(x,t) is the dynamic viscosity. The Dirac distribution function δs expresses the fact that the surface tension term is concentrated on the interface; σ is the surface tension coefficient, κ and n are the curvature and normal to the interface. Inside the domain existence of different fluids is captured using volume fraction c(x,t), which can be used for defining density and viscosity of the generalized fluid at a particular location as ρ(c )̃ = c ρ̃ 1 + (1 − c )̃ ρ2

(4)

μ(c )̃ = c μ̃ 1 + (1 − c )̃ μ2

(5)

Here subscripts 1 and 2 are used for continuous and discrete phases, respectively. Filtered variable c̃ in a cell is constructed by averaging eight neighboring values of volume fraction c(x,t) of a cell. Bilinear interpolation from cell centered values of leaf cells, having lower level, is used for finding the neighbors of the same level. Constitutive equation for volume fraction can be developed from eq 2 in the following form:

the domain in which the azimuthal (θ) direction is shown with respect to the roller. Incompressible Newtonian fluid pairs are taken as working fluid, and the corresponding properties are mentioned in Table 1. To explain the dynamics of wrapping liquid film and the formation of cusp by entrainment of one fluid into other and finally the liquid properties, we have considered two nondimensional numbers, namely, capillary πμrN 30σ

∂ρ + ∇·(ρu) = 0 ∂t

∇·u = 0

Figure 1. Computational domain used for simulation.

(

(1)

In the bulk where fluid density remains constant, eq 2 can be modified into the following form:

2. NUMERICAL METHODOLOGY 2.1. Computational Domain of the Problem. A 2D, 50 × 50 cm2, wall bounded computational domain is considered to study the entrainment dynamics around a 6 cm diameter, infinitely long horizontal roller. Figure 1 shows a schematic of

number Ca =

⎛ ∂u ⎞ ρ⎜ + u·∇u⎟ = −∇p + μ∇2 u + ρg + σκδsn ⎝ ∂t ⎠

∂c + ∇·(cu) = 0 ∂t

(6)

Finite volume based freeware Gerris is used for mesh generation and solution of discretized governing equations. Gerris discretizes the working domain using adaptable square boxes in 2D. Those elemental volumes are organized in hierarchic manner like quadtree. In zero level one square box breaks into four children cells and again all the children cells behave like

4

)) and Morton number (Mo = ρσgμ ), where 3

r is roller radius and N is roller speed in rpm. Generally, capillary Table 1. Fluid Properties Used for Simulation fluid pair pair 1 pair 2 pair 3 pair 4

name of the fluid water air glycerol air transformer oil air lubricating oil air

ρ (kg/m3)

μ (Pa·s) −3

1 × 10 1.789 × 0.023 1.789 × 0.031 1.789 × 0.136 1.789 ×

1000 1.225 1177.7 1.225 872.5 1.225 876.4 1.225 11951

Mo =

0.0720

2.63 × 10

0.0660

8.11 × 10−6

0.0320

3.17 × 10−4

0.0350

8.93 × 10−2

10−5 10

−5

10−5 10

−5

μ4 g

σ (N/m)

ρσ 3 −11

DOI: 10.1021/acs.iecr.6b03477 Ind. Eng. Chem. Res. 2016, 55, 11950−11960

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Industrial & Engineering Chemistry Research

Figure 2. Adaptive cells near interface and solid: (a) close-up view of grids near interface and solid; (b) complete computational domain discretized with grids in adaptive way.

Implicit Crank−Nicholson scheme is used for time discretization in multilevel poisson solver which is unconditionally stable.

parent cell and each cell breaks into another four cells, if required, in the next level. All nonparent cells are treated as 2 finite volume surrounded by neighbors. It generates 2n cells without adaptation, where n is the level. Figure 2 shows the mesh structure inside the domain, and one can notice finer meshes along the interface between fluids. Discretization of eqs 1, 2, and 6 is performed in staggered spatial grid and second order accurate scheme in time. Classical time splitting projection method17 is incorporated in Gerris to tackle the interface configurations for large density ratio of fluids. The equations take the following form of expressions:

1

to cater Courant number criterion. Changes of interface posi∂Δ tions with respect to time ∂t are kept less than 10−2 mm/s

( )

for obtaining steady state. 2.3. Grid Independence Test and Validation. At first, grid independence test has been performed in order to find the optimum refinement level of cells required to simulate our problem. For this purpose, simulations have been performed around a solid roller having diameter 6 cm and rotating across the interface of lubricating oil−air fluid pair (fourth pair in Table 1) with roller speed of 200 rpm. At different maximum refinement level, wrapping film thickness at different angular position is determined and reported in Table 2. To make a

⎡ u − ui ⎤ + ui + 1 ·∇ui + 1 ⎥ ρi + 1 ⎢ * 2 2⎦ 2⎣ Δt = −∇pi + 1 + ∇·[μi + 1 (Di + D )] + (σκδsn)i + 1 * 2 2 2

ci + 12 − ci − 12 Δt

+ ∇·(ciui) = 0

Δt ∇p ui + 1 = u − * ρ 1 i + 12 i+

( |Δu| )

Variable time step for marching is bounded by Δt < 2 min

(7)

Table 2. Wrapping Liquid Film Thickness (h) at Different Angular Position (θ) and Computational Time (s) with Different Refinement Level

(8)

(9)

refinement level

h (cm) at θ = 30°

h (cm) at θ = 60°

h (cm) at θ = 90°

computational time (s)

∇·ui + 1 = 0

(10)

⎡ ⎤ Δt ⎢ ∇· ∇p 1 ⎥ = ∇·u * ⎢⎣ ρi + 1 i + 2 ⎥⎦ 2

10 11 12

0.587762 0.615417 0.617136

0.290684 0.321106 0.323075

0.177935 0.205128 0.207302

8735.6 37646.7 295688.6

(11)

2

better optimized balance between accuracy and computational power, we have observed the wrapping liquid film thickness around the roller and the computational time (time taken to achieve steady state configuration) for different refinement level. We have calculated the film thickness at three different angular positions (30°, 60°, and 90°) at steady state configuration and observed the variation of accuracy level and also found out the time required to achieve the steady state situation for those refinement levels. It has been found that the change in wrapping film thickness is changing ∼1.05% in the cases of levels 11 and 12, but it is around 15.30% in the cases of levels 10 and 11. But level 12 consumes very high (685.43% increase compared with that of level 11) computational power for ∼1.05% change in film thickness. Hence, refinement level 11 has been chosen for our domain discretization in this paper. For validation of our numerical code, azimuthal film thickness around a roller (diameter 15 cm) rotating at 20 rad/s in air− ink interface (ρair = 1.225 kg/m3, μair = 1.789 × 10−5 Pa·s ρink = 1560 kg/m3, μink = 0.136 Pa·s, σ = 0.035 N/m) has been compared with observations of Yu et al. (2009). The roller is equally submerged in both the fluids and rotating in clock wise direction.

Here, D is the deformation tensor and can be expressed as ∂uj ⎞⎤ 1 ⎡⎛ ∂u Dij = 2 ⎢⎜ ∂xi + ∂x ⎟⎥ i ⎠⎦ ⎣⎝ j After calculation of the volume fraction c(x,t) from the discretized equations mentioned above, interface capturing needs spatial reconstruction, geometric flux, and interface advection computations. Gerris uses polynomial line m·x = α scheme for representation of linear interface in each cell. Here m is the local normal to the interface. x is the position vector of the cell, and α can be adjusted based on the volume fraction values of the cells lying below the slope of the proposed interface. For further detail of the interface reconstruction scheme one can refer to refs 18 and 19. Roller has been entered into the domain as an infinite long cylindrical solid with circumferential velocity μ = (μi,μj) = (ωr sin θ, ωr cos θ). Cells containing solid fractions are treated as mixed cell. Solid boundary is defined through a volume-of-fluid type approach devoid of surface tension force. Similar meshing algorithm and adaptations are used in both interface and solid boundary (Figure 2). 11952


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transient dynamics of wrapping and entrainment zone at 200 rpm for air−water fluid pair. During clockwise rotation of cylinder we have observed deviation of horizontal interface in both advancing and receding junction in a progressive (with time) manner. The interface reconstruction at steady state configuration in advancing (entrainment) and receding (wrap) section of roller has been shown in Figure 4 for air−water fluid pair, and the inset figures of Figure 4 show the temporal variation of t interfaces with nondimensionalized time, t * = τ (τ is the time required to achieve the steady state configuration, and t is actual time) before reaching the steady state. When t* = 1, then the steady state situation has been achieved; i.e., beyond that, the contour variation of phases has not been observed. Static contact angle of liquids with the solid surface has been considered as 90° for this simulation. The same has been considered for all simulations, presented in this article. Change of wettability for solid roller will alter the static contact angle and shift the location of the triple point. Present cases are specific for neutral liquid affinity surfaces, keeping the interface perpendicular to the solid surface at the triple line. It is to be noted that incorporation of dynamic contact angle at this point will produce more accurate results at contact line. In the advancing end, inertial force due to centrifugal motion, lighter fluid is pushed or entrained inside the heavier one. But due to higher density ratio, this penetration became very small, and due to buoyancy push, it diminishes quickly. In the advancing front, such entrainment and de-entrainment happen periodically to generate a wavelike structure in the free surface. In the lower inset of Figure 4, we explained this transition with nondimensionalized time (t*). In the receding end, heavier liquid, by virtue of upward component of centrifugal force, starts wrapping the cylinder. This wrap progressively thins down as vertical component of centrifugal force decreases with increase of θ. Horizontal component of centrifugal force keeps the liquid wrap adhered with the cylinder. During the

Excellent match between these two numerical simulations, as shown in Figure 3, gives confidence in our simulations.

Figure 3. Comparison of wrapping film thickness with the results of Yu et al.16 Adapted with permission from International Journal of Heat and Fluid Flow, Vol. 30, Yu, S. H.; Lee, K. S.; Yook, S. J., Film flow around a fast rotating roller, pp 796−803, Copyright 2009, Elsevier.

3. RESULTS AND DISCUSSION Present simulation develops an idea of liquid film wrapping over the infinite long horizontal rotating cylinder and entrainment of lighter fluid into the heavier one. Four different fluid pairs (gas−liquid) are chosen for the simulation with wide range of roller speed (0−200 rpm). Properties of the fluids are mentioned in Table 1. In this work we have not varied cylinder diameter (6 cm) and submergence ratio (0.5). Only clockwise rotation of the cylinder is considered as shown in Figure 1. At first, we present simulation results in terms of phase contour of air−water combination (Mo = 2.63 × 10−11). Figure 4 shows

Figure 4. Air−water (Mo = 2.63 × 10−11) interface due to rotation of cylinder at 200 rpm. Inset figure shows the temporal structures at advancing (entrainment) and receding (wrap) fronts. 11953


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Figure 5. Entrainment of air inside transformer oil (Mo = 3.17 × 10−4) at 200 rpm. Phase contours are plotted at steady state in main figure. Inset figures show transient sector coverage of wrap and entrainment at different rpm. Entry of wrap inside entrainment at steady state is also shown in inset.

Figure 6. Entrainment of air inside lubricating oil (Mo = 8.93 × 10−2) at 200 rpm. Inset figure shows entrainment rate at different rpm.

climb of liquid film, drainage also takes place which causes upward liquid flow inside the film near the cylinder and downward drainage away from the cylinder. This can be termed as double layer, and downward liquid drain creates an immediate crater in the liquid pool. Hence receding end is characterized by liquid wrap followed by a crater. Upward inset in Figure 4 shows the temporal growth of the both these features with nondimensionalized time. With increase of denser fluid viscosity or Mo, we observe that at steady state, extent of entrainment and wrap increases up to higher sector angle around the cylinder while compared at same

rotational speed. For demonstration we have shown steady state simulation results of transformer oil−air combination (pair 3 of Table 1; Mo = 3.17 × 10−4) at 200 rpm, in Figure 5. Upon comparison between steady state phase contours of Figures 4 and 5 (both at 200 rpm), we observe that at steady state, high viscosity liquid wraps the cylinder completely (θwrap ≥ 180°) whereas small sector angle (θwrap < 180°) is only wrapped for low viscosity liquid. Measurement of θwrap has been progressed from receding end in the clockwise direction. It is also noticed that steady wrap thickness decreases as θwrap increases around the cylinder. Shallower steady crater formation around double layer 11954


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Figure 7. Variation of steady contours of phases for different Mo or gas−liquid combination around the roller with angular speed of 200 rpm.

entrainment, buoyancy opposes vertical component of centrifugal force. In inset of Figure 5, transient behavior of both advancing entrainment and receding wrap have been characθwrap θ terized by nondimensional arc length (S* = entrain or 2π ) with 2π nondimensionalized time t*. It can be observed that with increase of rotational speed of the cylinder, at steady state more wrapping and entrainment can result. Steady nature of wrap and entrainment can be noticed from constant values of S* at t* ≥ 1 for receding and advancing fronts, respectively. It can be also commented that wrapping is stronger than entrainment as steady values of S* are higher in receding front than advancing. Above a critical speed of cylinder, at steady state, we observe

of air−transformer oil in comparison to air−water combination (Figure 4) has also been observed. This happens due to less penetration efficiency of drained liquid into higher viscosity pool. In the advancing front, extent of steady state entrainment also increases for viscous liquid (Figure 5) than less viscous one (Figure 4) and lighter air is seen to entrain up to higher subtended angle (θentrain) in air−transformer oil than air−water. Measurement of θentrain has been progressed from advancing end in the clockwise direction. But as vertical component of centrifugal force decreases with increase in θentrain, progressively thinner and steady entrained air layer is observed in the downward direction of the pool. In this zone (180° < θ < 270°) of 11955


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noticed entrainment volume not only grows with time but proportionately increases with rpm. It can be observed from Figures 4−6 that entrainment pattern and profile are a function of fluid properties. Nondimensional parameter Mo plays a major role in producing different structures during entrainment. Air−water combination (Mo = 2.63 × 10−11) is only able to create a crater in the advancing interface, whereas almost 90° wrapping of entrained film has been observed in the case of highly viscous lubricating oil−air combination (Mo = 8.93 × 10−2). To clearly assess the effect of the fluid properties, we have simulated entrainment patterns at the same rpm for a wide variety of fluid combinations as mentioned in Table 1. Figure 7 shows steady entrainment patterns for gas−liquid combinations with high buoyancy effect at 200 rpm. Mesh refinement near the entrained film is shown in inset which has a sufficient number of elements to probe the correctness of our prediction. Complete air film wrapping has not been observed in these cases, as opposing buoyancy force is of similar in magnitude as compared to the centrifugal one. With increase of Mo of the liquid we observed the penetration of entrained air at steady state increases when compared at the same rpm. At the receding front, increase of Mo (viscosity of liquid) makes the crater disappear. More clear pictures can be shown in Figure 8 which demonstrates azimuthal profile of nondimensional steady thickness of wrapping film (h/d) at the same rpm for three different gas−liquid combinations (Mo = 8.11 × 10−6, 3.17 × 10−4, and 8.93 × 10−2). First point is to be noted that in this range of Mo, at 0° angular position, the steady liquid film thickness is maximum because at that position the gravitational force is prompt. As we traverse in azimuthal plane for increasing θ, the influence of tangential gravitational force decreases due to which h gradually thins down. But the rate of decrease of h with respect to θ is more between 0° and 60°; beyond this position h is almost constant. One can observe film thickness increases with increase of Mo (liquid viscosity), and faster stabilization of liquid wrap thickness occurs for lower viscous glycerol−air combination (Mo = 8.11 × 10−6). For a particular roller speed, momentum transfer to the liquid film is higher for higher Mo liquid which helps the liquid film to grow thicker. Here observations are only shown at 200 rpm, but similar results have been observed for other rpm.

Figure 8. Variation of steady liquid wrap thickness for different gas− liquid pairs (Mo = 8.11 × 10−6, 3.17 × 10−4, and 8.93 × 10−2) at roller speed of 200 rpm. d is roller diameter. 1

total liquid wrapping (S* = 2 ) and even liquid wrap penetrate inside air entrainment attached with the solid. In those situations, one can observe a steady lighter air entrainment in the heavier liquid away from the cylinder. Extension of the liquid wrap inside air entrainment will cause a thin layer of heavier liquid between cylinder and air entrainment. This can lead to uniform steady wrap of heavier density fluid around the cylinder at subsequent higher rpm. Values of S* more than 0.5 confirm our observation at receding front. With increase of viscosity of the liquid or Mo further at higher rpm one can observe the steady air entrainment progresses beyond θ = 270° or θentrain = 90°. In this zone horizontal component of centrifugal force pushes the entrained air away from the cylinder. Once the entrained air detaches from the rotating solid, surface tension pinches it off and forms disconnected bubbles, which mainly by virtue of buoyancy diminish at the interface. This process continues a repeated number of times keeping a steady wrapping of air around the cylinder. As repeated pinch off cycles continues, in these cases, actual steady state of interface was never reached. Similar pseudo steady situation has been shown in Figure 6 for airlubricating oil (Mo = 8.93 × 10−2) combination at 200 rpm. In the inset of this figure we have also shown temporal variation of air accumulation inside liquid for different rpm. It can be

Figure 9. Effect of wrapping film thickness for different Mo with roller speed. 11956


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Figure 10. Velocity vectors around the roller for Mo = 3.17 × 10−4 at steady state. Enlarged view of receding and advancing front.

Figure 11. Azimuthal profile of dimensionless steady film thickness number (h*) with characteristic angular position (θ*). Mo has been varied in a wide range to obtain different gas−liquid combinations.

Figure 12. Comparison of dimensionless steady film thickness (h/d) with Ca for a wide range of Mo. Transformer oil−air (Mo = 3.17 × 10−4) and lubricating oil−air combination (Mo = 8.93 × 10−2) are shown here.

Azimuthal profile of steady wrapping liquid film over the solid roller at receding front is the most important parameter in this kind of study. Figure 9 shows the effect of Mo (liquid viscosity) on steady liquid film thickness for a wide range of roller speed. As Mo increases for the liquid−gas pair, film thickness at steady state increases. Here, we have obtained at least 3 times increment in thickness for the observed range of Mo. In the range of Mo observed, steady wrapping liquid film thickness is increasing with the roller speed. More rotational speed is associated with higher pull of air inside the liquid, and hence, increase of centrifugal force thickens the air entrainment. We also tried to observe the steady wrap film dynamics by observing the velocity vectors around the roller and specifically concentrate on advancing and receding front. Figure 10 shows the velocity vectors around the roller at 200 rpm for moderate Mo (transformer oil−air; Mo = 3.17 × 10−4). Receding front as shown in zoomed view clearly establishes two oppositely moving zones in wrapped film. Adjacent to the wall of the roller, liquid climbs clockwise due to centrifugal action, but in the periphery

Figure 13. Reconstruction of interface in steady state configuration for two gas−liquid pairs at 200 rpm. Transformer oil−air (Mo = 3.17 × 10−4) and lubricating oil−air combination (Mo = 8.93 × 10−2) are shown.

of the liquid wrap downward movement of velocity is observed. This essentially creates a situation of double layer. This also creates a circulation in the air zone at the receding end. 11957


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been found by balancing gravitational and viscous force as shown below: ρgh3 ≈ μUh h* =

or

h2 ≈

μU ρg

or

h≈

μU ρg

or

h μU ρg

(12)

We also define characteristic angular position (θ*) as

θ* =

θ − θhmax θhmin − θhmax

where θhmax and θhmin are the angular positions

at maximum and minimum values of liquid film thickness at steady state over the roller, respectively. In Figure 11, we have plotted the film thickness number with characteristic angular position for all three fluids at 200 rpm. It can be observed that azimuthal profile of film once nondimensionalized with viscous length follows a universal governing law as

Figure 14. Variation of steady cusp shape at different rotational speed (Ca) for two gas−liquid pairs. Transformer oil−air (Mo = 3.17 × 10−4) and lubricating oil−air combination (Mo = 8.93 × 10−2) are shown here.

h* = c1(N )θ*c2(N )

(13)

Value of c1(200) = 0.6 and c2(200) = 0.636. Similar values can be obtained for other rotational speed. But the important observation is that these nondimensional parameters are related by exponential eq 13. Capillary number (Ca) is the ratio of viscous force over surface tension forces which plays major role in wrapping/ entrainment dynamics. It has already been observed that at very low roller speed the liquid film is unable to give a complete wrap over the solid roller due to predominant action of surface tension on the wrapping liquid film. Figure 12 shows the variation of Ca with h* for Mo = 8.93 × 10−2 and 3.17 × 10−4 at different characteristic angular position (θ*). It is clear from the figure that the steady wrap thickness is more at lesser

While the liquid wrap is falling down, it bombards the liquid pool to make a crater or dimple in the free surface. This zone of crater decreases as Mo increases, and finally at very high Mo (lubricating oil) strength of falling liquid layer is not enough to create dimple in the pool. In the advancing front, an air jet is seen to be produced which helps the penetration of gas in liquid. In the free surface lots of undulations are being observed which sometimes accommodate the plunging jet inside the liquid as a cusp. An effort has also been made to scale the steady film thickness and obtain nondimensionalized thickness number h* to propose universal governing laws. Film thickness number has

Figure 15. Force balance for both liquid wrap and entrained air cusp for analytical prediction of film thickness. 11958


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Figure 16. Comparison between analytical formulation (eq 16) and numerical prediction for four different fluid pairs.

Mainly we observe two types of film around the cylinder, i.e., wrapping liquid film over the roller in advancing side and entrained air film in receding side of the roller. These liquid films experience different kinds of forces on them as shown in Figure 15. The respective directions of viscous, buoyancy, and centrifugal forces are shown in detail for simplicity. If we consider a small section of entrained air film, forces that are acting on it can be shown in terms of free body diagram. This section experiences centrifugal force in the tangential downward direction of roller at a point and viscous force just opposite to it. On the other hand, buoyancy force is in the vertically upward direction for this light air phase submerged in liquid. By balancing the forces, we find the following expression:

angular sector and it diminishes as angle increases. Increase of rotational speed or Ca also has a prompt effect on azimuthal film thickness. For all the fluid combinations, we have observed that the steady film increases its thickness as the speed of rotation picks up. This behavior of film is true for a wide range of Mo and Ca. Lubricating oil−air (Mo = 8.93 × 10−2) at the same speed has higher Ca than transformer oil−air combination (Mo = 3.17 × 10−4). In Figure 13, we have also compared the steady wrap thickness for both these combinations side by side at 200 rpm. This once again confirms our observation of Figure 12. Another interesting feature is that at higher Mo (viscosity), crater formation is not observed as shown in Figure 13. On the other hand, a study has also been made to characterize the air cusp entrainment angle at steady state along with the rotational speed. For two different gas−liquid combinations, in Figure 14, we showed that cusp entrainment angle increases with increase of rotational speed or Ca. The observation is valid for a wide range of Morton number. The reason behind this observation can be explained using the velocity vectors shown in Figure 10. In the advancing end, a jet of air imposed by centrifugal force causes the entrainment. With increase of rotational speed or Ca, the strength of the jet increases causing greater subtended angle by air cusp. After observing the contour of wrap or film, we tried to analytically observe the mutual interplay of influencing forces.

Fcentrifugal − Fviscous = Fbuoyancy cos θ

(14)

It can be further expressed in terms of liquid properties, rpm, and film thickness. ρa ω 2rh3 −

μa rω(1 − a) h

h2 = (ρa g cos θ )h3

(15)

Here, a is the fraction of centrifugal velocity being reduced due to viscous damping. Simplification of eq 15 leads to expression of film thickness as 11959


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Industrial & Engineering Chemistry Research h=

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μa rω(1 − a) ρa [ω 2r − g cos θ ]

(16)

Expression obtained for steady wrapped film thickness has been compared with numerical observations considering a = 0.2. In Figure 16, we have compared analytical prediction for all four combinations. The match between this simplified expression and computational result is quite satisfactory. Similar expression can also be obtained for steady wrapped film by balancing the forces on the upper element as shown in Figure 15. Accuracy of prediction in liquid wrap is similar, as has been observed for gas entrainment for different Mo.

4. CONCLUSIONS A numerical study has been performed in order to understand the gas entrainment and liquid wrap flow around the rotating roller in a stratified layer of gas−liquid pair, where the roller is equally submerged in both of the fluids. Gaseous cusp entrainment and its variation of thickness in azimuthal plane for different roller speed have been observed. In advancing front, we have observed cusp formation, air film subtended around cylinder, and entry of wrap inside entrainment depending on Mo and Ca. By nondimensionalizing by viscous length scale, we have shown that all fluid pairs entrain in a similar fashion and they can be explained by a unified relationship between thickness number and angle subtended. In the receding end, existence of double layer and crater formations is observed numerically. By showing the velocity vectors at the advancing and receding end, we explain the mechanisms of all interfacial behavior. Finally, by a free body force balance of air entrain strip and wrap strip, we have predicted the film thickness and compared results with numerical findings satisfactorily.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Numerical Study of Air Entrainment and Liquid Film Wrapping around

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