- Email: [email protected]
Modelling the behaviour of the wetting front in non-standard forced foam drainage scenarios
Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 21–27
Contents lists available at ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Modelling the behaviour of the wetting front in non-standard forced foam drainage scenarios P.R. Brito-Parada a,b,∗ , S.J. Neethling a , J.J. Cilliers a a Rio Tinto Centre for Advanced Mineral Recovery, Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom b Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
The dynamics of the wetting front in forced foam drainage are studied numerically. The model benefits from the use of adaptive, unstructured finite element meshing. The horizontal displacement of the wave also obeys a power law for localised cases. Forced drainage in non-rectangular containers is simulated for the first time. Different wetting front regions in non-rectangular cases also adjust to a power law.
a r t i c l e
i n f o
Article history: Received 13 November 2012 Received in revised form 9 February 2013 Accepted 12 February 2013 Available online 21 February 2013 Keywords: Foam drainage Forced drainage Finite element method Mathematical modelling
a b s t r a c t Forced foam drainage experiments, in which liquid is added at a constant rate at the top of the foam, are studied numerically. The aim of these experiments is to investigate the change in liquid fraction as the resulting drainage wave propagates through the system. A finite element implementation of the foam drainage equation is used to carry out two-dimensional simulations, taking advantage of mesh adaptivity techniques to accurately resolve the dynamics of the wetting front. First, the effects of changes in the liquid addition area at the top of a rectangular container are studied, showing that the variation of the position of the wave front exhibits a power law with time not only for the vertical displacement but also for the horizontal propagation. Then, for uniform addition scenarios, the effect of changes in the geometry of the container are analysed, finding that a power law also describes well the position of the different regions of the drainage wave with time. © 2013 Elsevier B.V. All rights reserved.
1. Introduction
∗ Corresponding author at: Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom. Tel.: +44 20 7594 7145. E-mail address: [email protected] (P.R. Brito-Parada). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.02.013
The drainage of liquid in two-phase foams is of importance to a number of industrial applications which depend upon the liquid fraction through the system. The behaviour of the liquid in the Plateau border network of interconnected channels is governed by gravity and capillary forces, with the process also being affected by viscous losses.
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Two limiting regimes are used to describe the process, depending on the dissipative structure considered to effect the viscous losses, namely the channel dominated and the node dominated drainage regimes [1]. Whilst the former, known as the standard drainage theory, considers the dissipation to take place in the Plateau border channels, the latter considers the viscous losses to occur in the nodes where four Plateau borders meet. Foam drainage experiments in which surfactant solution is added at a constant rate at the top of a foam, covering the entire surface width, have been referred to as forced drainage [1,2]. The aim in these experiments is to investigate the change in liquid content as the resulting drainage wave propagates through the system. Since liquid addition occurs over the entire top surface of the foam, the problem can be formulated in one dimensional form. This particular case played a key role in the development of an analytical solution to the standard drainage equation in the form of a solitary wave [3] that propagates at constant velocity throughout the height of the foam. Some variants of the classic forced drainage experiment have been carried out. Hutzler et al. [4] performed experiments and simulations in which the width of the addition input at the top of a rectangular Hele-Shaw cell was varied, the problem thus becoming two-dimensional. They found good agreement between experiments and simulations considering Poiseuille flow, and concluded that the vertical displacement of the wave front with time can be described by a power law, with the exponent varying with the width of the liquid addition input. The experimental and numerical data reported by Hutzler et al. [4] was obtained by tracking the wave at fixed positions, i.e. at the centre of the cell width for the vertical spreading and at a fixed height near the top for the horizontal position. Whilst the wave front in the vertical coincided with the measurement position, this did not seem to be the case for the horizontal spreading caused by capillary forces, for which a power law fit was not found to describe the data at the selected measuring position. A closer look at the actual maximum horizontal value of the wave front, which changes in height as drainage occurs, would be necessary to determine whether a scaling law holds. A similar experimental setup in which liquid was not added uniformly at the top of the foam was investigated by Huang and Sun [5]. In these experiments two liquid addition nozzles were employed, and the effect of their separation distance on the dynamics of the wave front was studied. The numerical modelling of the problem carried out by Huang and Sun [5] did not make use of the experimental values of liquid addition rate, but instead they fitted a boundary condition for the liquid fraction at the inputs, after running a number of simulations and comparing the results to the experiments. The use of source or flux terms would be a more adequate approach to better represent the liquid flowrate in these type of simulations. Other examples of two-dimensional numerical modelling of foam systems with liquid addition at the top can be found in simulations of froth washing in flotation cells and columns [6,7]. However these simulations looked at steady state results for a flowing foam, a situation even further away from the conditions of the static foam and dynamic wetting front of the standard forced drainage problem. In terms of container shape effects on liquid drainage, a special case can be found in the Eiffel tower experiments by Saint-Jalmes et al. [8]. In these free drainage experiments the shape of the vessel eliminates vertical wetness gradients, and a solution of a standard drainage equation modified to account for variable crosssectional area is presented for the aforementioned geometry. A mathematical description and modelling of the problem is therefore performed without the carrying out of fully two-dimensional simulations.
Previous numerical work has also focussed on the mathematical description of the boundary layer of wet foam that appears near the wall at sloping weirs, and the description of the liquid jet that travels down along the wall [9,10]. Grassia and Neethling [10] mention that one of the challenges to perform full numerical simulation of the drainage equation is the fact that a grid fine enough to resolve the boundary layer at the weir can be prohibitively expensive computationally. One would expect similar problems when numerically solving forced drainage scenarios in containers with sloping walls. However, whereas the above is true for uniformly sized structured grids, unstructured meshes and mesh adaptivity techniques could be used to more efficiently representing the domain, focussing resolution on boundary layers and resolving dynamic features such as wave fronts in greater detail than the rest of the domain. By considering the different variants of the forced drainage experiments, the problem could be categorised, for example, by the extent of the addition rate area at the top into uniform or localised forced drainage. It could also be classified by the shape of the vessel in which the foam is contained. In this paper, we carry out two dimensional simulations of two types of non standard forced drainage setups: localised forced drainage with increasing input width in a rectangular container and, for the first time, uniform forced drainage for non-rectangular containers. 2. Foam drainage model and numerical techniques All the simulations in this work make use of a finite element implementation of the standard foam drainage equation, thus following channel dominated drainage theory. Fluidity [11], a fluid dynamics code that uses unstructured finite element meshes, has been used as the modelling framework. As in previous implementations [12,13], our model considers gravity, g, capillarity and viscous dissipation to be the forces determining the velocity of liquid, u, through the Plateau borders of cross-sectional area A. The balance of the aforementioned forces for a liquid of density , surface tension and viscosity , yields −g −
C −3/2 3C u ∇ A − PB A = 0, 2 A
(1)
where C results from geometrical considerations of the Plateau borders cross sectional area [12]: C=
√ 3− , 2
and CPB is the drag coefficient, a factor based on the shape of the Plateau border channels, multiplied in Eq. (1) by a factor of 3 to account for the random orientation of the Plateau borders [12]. Since the only liquid being considered is that within the Plateau borders, the liquid fraction in the foam, ε, can be defined as the ratio of the volume of liquid in the Plateau borders and the volume of the system. Following [13], in order to convert the balance from being over a single Plateau border to being over a volume of foam containing many of these channels, the variable is defined as the local Plateau border length per unit volume, which results in ε = A.
(2)
The length of the Plateau borders per unit volume can be determined for foam cells of equivalent sphere radius R. The model in this paper makes use of a semi-empirical relation obtained from simulations of foams with random structure [14] that leads to =
3.3 R
4 3
R3
−1/3
4
+ 0.063
3
R3
−2/3
.
P.R. Brito-Parada et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 21–27
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Assuming that the density of the liquid is constant, the conservation of liquid can be expressed as
∂ε + ∇ · (εu) = 0. ∂t
(3)
The solution of Eqs. (1)–(3) provides a description of the liquid velocity, the Plateau borders cross-sectional area and the liquid fraction through the foam. The classic form of the standard foam drainage equation is easily obtained from substitution of the values for u and ε from Eqs. (1) and (2), respectively, into Eq. (3). In terms of boundary conditions, the bubbles at the liquid-foam interface can be considered to be nearly spherical and for randomly packed bubbles liquid fraction takes a value of approximately 0.36. For the walls and top surface the boundary conditions for liquid velocity can be expressed as no normal flow. In Fluidity, the Galerkin finite element method [15] is used to approximate the solution of the continuous foam drainage model by transforming the equations to a weak form, multiplying them by a test function and integrating over the domain. Rather than looking for a solution of the fields in the entire function space, a linear combination of basis functions is used. For the simulations in this work we use piecewise linear basis functions for the liquid fraction and the Plateau border cross-sectional area, and discontinuous piecewise constant basis functions for the liquid velocity field. A detailed description of the model and the reasons behind the selection of these particular basis functions for the fields in the foam drainage equation will be the subject of a separate publication. Our model also takes advantage of the anisotropic adaptive unstructured meshing techniques implemented in Fluidity [16,17]. The size and shape of the elements in the mesh can thus vary in order to dynamically resolve developing solution features. Adaptive mesh simulations of gravity current fronts, for example, have been shown to perform as well as high resolution fixed mesh simulations while using at least one order of magnitude fewer nodes and with a minimal cost for the mesh adapting [18]. Similarly, details of a performance analysis for simulations of liquid drainage in an upward flowing foam using Fluidity’s adaptive remeshing techniques can be found in [19]. The accurate tracking of the wetting fronts that develop in forced drainage experiments would also be greatly benefited from the use of anisotropic adaptive remeshing. Fig. 1 shows and example of a uniform forced drainage simulation that takes advantage of this technique.
Fig. 1. Simulation results for a uniform forced drainage experiment showing the liquid fraction (left) and the corresponding adapted mesh (right) at different simulation times. As the wave front propagates downwards, the size of the elements in the mesh is reduced in the vicinity of the wave front in order to resolve the dynamics.
3. Localised forced drainage simulations Contrary to standard uniform forced drainage, in which gravity effects are dominant, localised forced drainage experiments allow further investigation of the effect of capillary forces on the horizontal spreading of liquid in the foam. Localised forced drainage simulations were performed for a rectangular Hele-Shaw cell of width 12 cm and depth 0.3 cm. The domain is the same as the one studied in [4], although a wider range of widths for the liquid addition input, from 0.3 cm to 6 cm, were analysed in this work. A monodisperse foam was considered, with bubble radius R = 5 ×10−2 cm. The initial liquid fraction for all the simulations was set to 5 × 10−3 , and the liquid addition at the top of the foam was achieved by considering a source term for the liquid content. As in previous localised forced drainage experiments [4], the liquid addition flowrate per unit input width was kept the same as the width was varied. The flowrate of liquid addition for the smallest input width was 5 × 10−3 cm3 s−1 . For the simulations in this work the physical parameters were set as follows: = 1 ×103 kg m−3 , = 2.5 × 10−2 N m−1 and = 1 ×10−3 Pa s. The criteria employed for mesh adaptivity in these simulations is based on an interpolation error for liquid fraction, set to 5 × 10−4 . The edge length of the elements in the mesh was allowed to vary
with upper and lower limits of 5 × 10−1 cm and 5 × 10−3 cm, respectively. For further details on the mesh adaptivity procedure in Fluidity see [11,16]. Fig. 2 shows the simulation results for liquid fraction at time t = 10 s for input widths of 0.3, 0.6, 1.5, 3, 4.5 and 6 cm. The vertical position of the wave front was measured at the centre of the input addition zone. Results obtained by tracking the wave front for a liquid fraction ε = 0.02, can be seen for the different input widths in a double logarithmic plot in Fig. 3. The data adjust to a power law, with the behaviour of the front in the vertical direction being practically the same above an input width of 3 cm. For the horizontal spreading, rather than considering the position at a constant height as in previous experiments [4], the wave front was tracked at the position corresponding to the maximum horizontal value of the liquid fraction isocontour, which changes in height with time. Results for the horizontal position at different input widths are presented in Fig. 4. We found the data to be well described by a power law for t ≥ 4 s. The values of the exponent in the power law describing the position of the wave front in time are shown in Fig. 5 for the different input widths and for both the vertical and horizontal spreading. These values have been obtained from data at t ≥ 4 s. Since the
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. . . . . . . . Fig. 4. Horizontal position (from centre of the input width) of the wave front in the localised forced drainage simulations; the maximum horizontal value of the liquid fraction isocontour ε = 0.02 is tracked. Different symbols correspond to different input widths.
exponents are dependent on the isocontours of liquid fraction being tracked, we show the values for ε = 0.02, corresponding to the results in Figs. 3 and 4, as well as the values for isocontours of ε = 0.006 and ε = 0.05. The exponent for the vertical spreading increases with input width, in agreement with the experiments and simulations by Hutzler et al. [4]. This increase, however, is only marginal above an input width of 3 cm. On the other hand, as the width of the localised input increases the exponent in the power law characterisation of the horizontal position with time decreases. The data show a faster displacement in the vertical direction if the maximum isocontour value is tracked. For the horizontal position
Fig. 2. Liquid fraction numerical results at 10 s for input widths of: (a) 0.3 cm, (b) 0.6 cm, (c) 1.5 cm, (d) 3 cm, (e) 4.5 cm, and (f) 6 cm.
0.02
0.05
Power law exponent - vertical
0.006
. . . . . . . .
Fig. 3. Vertical position (from top) of the wave front in the localised forced drainage simulations; values are for an isocontour of liquid fraction ε = 0.02 and different symbols correspond to different input widths.
Power law exponent - horizontal
Input width (cm)
Input width (cm)
Fig. 5. Exponents in the power law fit for the horizontal and vertical position of the wetting front in the localised forced drainage experiments at different input widths.
P.R. Brito-Parada et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 21–27
.
25
. . . . . . .
Fig. 6. Horizontal displacement (measured from the closest edge of the liquid addition zone) of the wave front in the localised forced drainage simulations; the maximum horizontal value of the liquid fraction isocontour ε = 0.02 is tracked. Different symbols correspond to different input widths.
the isocontour corresponding to the lowest liquid fraction value presents a faster displacement. The results for the horizontal position of the wetting front described above considered the distance of the front from the centre of the domain. If instead the actual displacement of the wetting front is tracked, i.e. that from the closest edge of the liquid addition zone, it becomes clearer that the data for the horizontal displacement of the front can be well described by a power law, as seen from the double logarithmic plot in Fig. 6. The exponent in the power law describing the horizontal displacement, as defined above, increases with input width only initially, with its value not varying significantly above an input width of 1.5 cm. This is shown in Fig. 7, in which values are presented for isocontours of different liquid fractions. Although the setup in these numerical experiments was similar to that presented in [4], a quantitative comparison of results would require information not reported in the aforementioned work (in particular the physical properties of the liquid in the simulations). However, we found qualitative agreement between the results for the vertical position of the front with regards to its power law behaviour.
Fig. 7. Exponents in the power law fit for the horizontal displacement of the wetting front in the localised forced drainage experiments at different input widths.
We have shown that the horizontal displacement of the wave front in these type of experiments is also well described by a power law when tracking the maximum horizontal value of the front and not that at a fixed height as in previous studies [4]. This is particularly clear if the actual displacement is analysed, and not the position with respect to the centre of the liquid addition zone. Our results also indicate an initial increase in the exponent characterising the power law for both vertical and horizontal displacement, but only up to a certain input width, above which the value of the exponent remains practically the same. The input width at which this is observed is, however, different for the vertical and horizontal displacements. 4. Uniform forced drainage simulations in non-rectangular containers Another variation of the classic forced drainage experiment arises if non-rectangular containers are considered. By changing the shape of the container different liquid fraction profiles can be obtained, which results in deviations from a uniform wetting front that moves with constant velocity. In this section we study the effect of four different container shapes on the dynamics of the wetting front when liquid is added all across the top of the foam.
Fig. 8. Non-rectangular container shapes considered for the forced drainage simulations: (a) converging planar walls; (b) diverging planar walls; (c) converging curved walls; (d) diverging curved walls.
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Fig. 9. Liquid fraction numerical results at 50 s, showing an isocontour of ε = 0.02, for containers with: (a) converging planar walls; (b) converging curved walls; (c) diverging planar walls; (d) diverging curved walls.
Vertical position (cm)
10 Converging curved-advanced front Converging planar-advanced front Converging curved-centre Converging planar-centre
1
0.1 1
10 Time (s)
100
Fig. 10. Positions of the wave front in the forced drainage simulations with converging walls; values are for an isocontour of liquid fraction ε = 0.02.
10
Vertical position (cm)
Simulations of uniform forced drainage experiments are carried out for Hele-Shaw type cells (5 cm in height and 0.3 cm in depth) with converging and diverging walls; for each case planar and curved walls are also evaluated. The geometries and dimensions of the four domains are shown in Fig. 8. The same physical properties for the liquid, bubble size, initial liquid fraction and addition flowrate per unit input width as in Section 3 were considered. Mesh adaptivity was also used in these simulations, with the same criteria employed for the localised forced drainage cases in the previous section. For clarity, the analysis of the results from the simulations are presented first for the containers with converging walls, followed by those with diverging walls. This is due to the similarities that the developing wetting fronts present. The liquid fraction profiles at 50 s for the four different containers are presented in Fig. 9, including an isocontour of ε = 0.02 for each case. It is observed that in containers with converging walls a high liquid fraction region develops at the walls, which travels down the foam at a higher velocity than the centre of the wave front. We refer to this high speed liquid jet as the advanced front. For containers with diverging walls, on the other hand, it is seen that the fastest region of the wave front is that at the centre of the container width, whilst at the walls a delayed front develops. Fig. 10 shows, in a double logarithmic graph, the vertical position of the wetting front for the containers with converging walls for t ≥ 4 s. The data can be well described by a power law, with different regions of the wetting front presenting differences in behaviour. The wetting front centre propagates at the same speed regardless of the presence of planar or curved walls. Curved converging walls result in an advanced front at the walls whose displacement is initially greater than the one for the converging planar walls, although it can be seen from Fig. 10 that the latter exhibits a higher power law exponent. The results for the position of the wetting front when using diverging walls are presented in Fig. 11 for t ≥ 4 s. As before, a power law behaviour was found. The displacement of the front at the centre, which is the fastest moving region for these configurations, is almost the same for both planar and curved walls. For the container geometries presented, the displacement of the delayed front in the case of diverging planar walls is greater than the one for diverging curved walls. Although this study considers only a few variations in container geometry, the finding that forced drainage in non-rectangular vessels can be well represented by a power law for the different regions
Diverging planar-centre Diverging curved-centre Diverging planar-delayed front Diverging curved-delayed front
1
0.1 1
10 Time (s)
100
Fig. 11. Positions of the wave front in the forced drainage simulations with diverging walls; values are for an isocontour of liquid fraction ε = 0.02.
P.R. Brito-Parada et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 21–27
is encouraging for the carrying out of experimental and further numerical work. 5. Conclusions Simulations of localised forced drainage scenarios in rectangular containers have shown that the position of the wetting front with time is well described by a power law not only for the vertical spreading but also for the horizontal displacement due to capillary forces. The data also indicate that increasing the input width results in an initial increase in the exponent for the power law describing the displacement, which is followed by a plateau. The input width above which the value of the exponent stops changing significantly was found to be different for the vertical and horizontal displacements. Numerical investigations of uniform forced drainage experiments were carried out for non-rectangular containers. It was found that the position of the distinctive regions of the wetting front also obeys a power law behaviour. For containers with converging walls a high liquid fraction zone develops at the walls and moves faster than the centre of the wetting front. In the presence of diverging walls it is the centre of the front that moves faster, as the horizontal transport of liquid to lower liquid fraction regions due to the shape of the walls results in a delayed section of the wetting front. The displacement of the centre of the front was found not to be affected by the planar or curved nature of the walls for either converging or diverging wall configurations. The finite element implementation of the foam drainage equation used in this work makes use of adaptive remeshing techniques to capture the dynamics of the wetting front. Our model can be used to numerically study forced drainage for arbitrary container geometries. Future research directions include further analysis of the wetting front in localised forced drainage experiments, including the behaviour once the wetting front reaches the walls of the container, as well as exploring the effects of flowrate variations in non-standard forced drainage scenarios. Three-dimensional forced drainage simulations will also be performed in future work.
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References [1] P.M. Kruglyakov, S.I. Karakashev, A.V. Nguyen, N.G. Vilkova, Foam drainage, Curr. Opin. Colloid Interface Sci. 13 (3) (2008) 163–170. [2] S.A. Koehler, S. Hilgenfeldt, H.A. Stone, A generalized view of foam drainage: experiment and theory, Langmuir 16 (2000) 6327–6341. [3] G. Verbist, D. Weaire, A soluble model for foam drainage, Europhys. Lett. 26 (8) (1994) 631–634. [4] S. Hutzler, S.J. Cox, G. Wang, Foam drainage in two dimensions, Colloids Surf. A 263 (2005) 178–183. [5] J. Huang, Q. Sun, Foam drainage with twin inputs in two dimensions, Colloids Surf. A 309 (2007) 132–136. [6] S.J. Neethling, J.J. Cilliers, Simulation of the effect of froth washing on flotation performance, Chem. Eng. Sci. 56 (21-22) (2001) 6303–6311. [7] S.J. Neethling, K. Hadler, J.J. Cilliers, A.W. Stradling, The use of FrothSim to optimise the water addition to a column flotation cell, Miner. Eng. 19 (6-8) (2006) 816–823. [8] A. Saint-Jalmes, M.U. Vera, D.J. Durian, Free drainage of aqueous foams: container shape effects on capillarity and vertical gradients, Europhys. Lett. 50 (5) (2000) 695–701. [9] P. Grassia, S.J. Neethling, J.J. Cilliers, Foam drainage on a sloping weir, Eur. Phys. J. E 8 (2002) 517–529. [10] P. Grassia, S.J. Neethling, Iterative approach to weir drainage, Chem. Eng. Sci. 59 (20) (2004) 4349–4359. [11] AMCG, Fluidity 4.1 Manual, Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, 2011 http://amcg.ese.ic.ac.uk/Fluidity [12] G. Verbist, D. Weaire, A.M. Kraynik, The foam drainage equation, J. Phys.: Conden. Matter 8 (21) (1996) 3715–3731. [13] S.J. Neethling, J.J. Cilliers, E.T. Woodburn, Prediction of the water distribution in a flowing foam, Chem. Eng. Sci. 55 (19) (2000) 4021–4028. [14] A.M. Kraynik, D.A. Reinelt, F. van Swol, Structure of random foam, Phys. Rev. Lett. 93 (20) (2004), 208301-1-208301-4. [15] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th ed., Elsevier Butterworth-Heinemann, Oxford, 2005. [16] C.C. Pain, A.P. Umpleby, C.R.E. de Oliveira, A.J.H. Goddard, Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Meth. Appl. Mech. Eng. 190 (29–30) (2001) 3771–3796. [17] M.D. Piggott, P.E. Farrel, C.R. Wilson, G.J. Gorman, C.C. Pain, Anisotropic mesh adaptivity for multi-scale ocean modelling, Philos. Trans. R. Soc. A 367 (1907) (2009) 4591–4611. [18] H.R. Hiester, M.D. Piggott, P.A. Allison, The impact of mesh adaptivity on the gravity current front speed in a two-dimensional lock exchange, Ocean Modell. 38 (1-2) (2011) 1–21. [19] P.R. Brito-Parada, S.J. Neethling, J.J. Cilliers, The advantages of using mesh adaptivity when modelling the drainage of liquid in froths, Miner. Eng. 33 (2012) 80–86.
Contents lists available at ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Modelling the behaviour of the wetting front in non-standard forced foam drainage scenarios P.R. Brito-Parada a,b,∗ , S.J. Neethling a , J.J. Cilliers a a Rio Tinto Centre for Advanced Mineral Recovery, Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom b Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
The dynamics of the wetting front in forced foam drainage are studied numerically. The model benefits from the use of adaptive, unstructured finite element meshing. The horizontal displacement of the wave also obeys a power law for localised cases. Forced drainage in non-rectangular containers is simulated for the first time. Different wetting front regions in non-rectangular cases also adjust to a power law.
a r t i c l e
i n f o
Article history: Received 13 November 2012 Received in revised form 9 February 2013 Accepted 12 February 2013 Available online 21 February 2013 Keywords: Foam drainage Forced drainage Finite element method Mathematical modelling
a b s t r a c t Forced foam drainage experiments, in which liquid is added at a constant rate at the top of the foam, are studied numerically. The aim of these experiments is to investigate the change in liquid fraction as the resulting drainage wave propagates through the system. A finite element implementation of the foam drainage equation is used to carry out two-dimensional simulations, taking advantage of mesh adaptivity techniques to accurately resolve the dynamics of the wetting front. First, the effects of changes in the liquid addition area at the top of a rectangular container are studied, showing that the variation of the position of the wave front exhibits a power law with time not only for the vertical displacement but also for the horizontal propagation. Then, for uniform addition scenarios, the effect of changes in the geometry of the container are analysed, finding that a power law also describes well the position of the different regions of the drainage wave with time. © 2013 Elsevier B.V. All rights reserved.
1. Introduction
∗ Corresponding author at: Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom. Tel.: +44 20 7594 7145. E-mail address: [email protected] (P.R. Brito-Parada). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.02.013
The drainage of liquid in two-phase foams is of importance to a number of industrial applications which depend upon the liquid fraction through the system. The behaviour of the liquid in the Plateau border network of interconnected channels is governed by gravity and capillary forces, with the process also being affected by viscous losses.
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Two limiting regimes are used to describe the process, depending on the dissipative structure considered to effect the viscous losses, namely the channel dominated and the node dominated drainage regimes [1]. Whilst the former, known as the standard drainage theory, considers the dissipation to take place in the Plateau border channels, the latter considers the viscous losses to occur in the nodes where four Plateau borders meet. Foam drainage experiments in which surfactant solution is added at a constant rate at the top of a foam, covering the entire surface width, have been referred to as forced drainage [1,2]. The aim in these experiments is to investigate the change in liquid content as the resulting drainage wave propagates through the system. Since liquid addition occurs over the entire top surface of the foam, the problem can be formulated in one dimensional form. This particular case played a key role in the development of an analytical solution to the standard drainage equation in the form of a solitary wave [3] that propagates at constant velocity throughout the height of the foam. Some variants of the classic forced drainage experiment have been carried out. Hutzler et al. [4] performed experiments and simulations in which the width of the addition input at the top of a rectangular Hele-Shaw cell was varied, the problem thus becoming two-dimensional. They found good agreement between experiments and simulations considering Poiseuille flow, and concluded that the vertical displacement of the wave front with time can be described by a power law, with the exponent varying with the width of the liquid addition input. The experimental and numerical data reported by Hutzler et al. [4] was obtained by tracking the wave at fixed positions, i.e. at the centre of the cell width for the vertical spreading and at a fixed height near the top for the horizontal position. Whilst the wave front in the vertical coincided with the measurement position, this did not seem to be the case for the horizontal spreading caused by capillary forces, for which a power law fit was not found to describe the data at the selected measuring position. A closer look at the actual maximum horizontal value of the wave front, which changes in height as drainage occurs, would be necessary to determine whether a scaling law holds. A similar experimental setup in which liquid was not added uniformly at the top of the foam was investigated by Huang and Sun [5]. In these experiments two liquid addition nozzles were employed, and the effect of their separation distance on the dynamics of the wave front was studied. The numerical modelling of the problem carried out by Huang and Sun [5] did not make use of the experimental values of liquid addition rate, but instead they fitted a boundary condition for the liquid fraction at the inputs, after running a number of simulations and comparing the results to the experiments. The use of source or flux terms would be a more adequate approach to better represent the liquid flowrate in these type of simulations. Other examples of two-dimensional numerical modelling of foam systems with liquid addition at the top can be found in simulations of froth washing in flotation cells and columns [6,7]. However these simulations looked at steady state results for a flowing foam, a situation even further away from the conditions of the static foam and dynamic wetting front of the standard forced drainage problem. In terms of container shape effects on liquid drainage, a special case can be found in the Eiffel tower experiments by Saint-Jalmes et al. [8]. In these free drainage experiments the shape of the vessel eliminates vertical wetness gradients, and a solution of a standard drainage equation modified to account for variable crosssectional area is presented for the aforementioned geometry. A mathematical description and modelling of the problem is therefore performed without the carrying out of fully two-dimensional simulations.
Previous numerical work has also focussed on the mathematical description of the boundary layer of wet foam that appears near the wall at sloping weirs, and the description of the liquid jet that travels down along the wall [9,10]. Grassia and Neethling [10] mention that one of the challenges to perform full numerical simulation of the drainage equation is the fact that a grid fine enough to resolve the boundary layer at the weir can be prohibitively expensive computationally. One would expect similar problems when numerically solving forced drainage scenarios in containers with sloping walls. However, whereas the above is true for uniformly sized structured grids, unstructured meshes and mesh adaptivity techniques could be used to more efficiently representing the domain, focussing resolution on boundary layers and resolving dynamic features such as wave fronts in greater detail than the rest of the domain. By considering the different variants of the forced drainage experiments, the problem could be categorised, for example, by the extent of the addition rate area at the top into uniform or localised forced drainage. It could also be classified by the shape of the vessel in which the foam is contained. In this paper, we carry out two dimensional simulations of two types of non standard forced drainage setups: localised forced drainage with increasing input width in a rectangular container and, for the first time, uniform forced drainage for non-rectangular containers. 2. Foam drainage model and numerical techniques All the simulations in this work make use of a finite element implementation of the standard foam drainage equation, thus following channel dominated drainage theory. Fluidity [11], a fluid dynamics code that uses unstructured finite element meshes, has been used as the modelling framework. As in previous implementations [12,13], our model considers gravity, g, capillarity and viscous dissipation to be the forces determining the velocity of liquid, u, through the Plateau borders of cross-sectional area A. The balance of the aforementioned forces for a liquid of density , surface tension and viscosity , yields −g −
C −3/2 3C u ∇ A − PB A = 0, 2 A
(1)
where C results from geometrical considerations of the Plateau borders cross sectional area [12]: C=
√ 3− , 2
and CPB is the drag coefficient, a factor based on the shape of the Plateau border channels, multiplied in Eq. (1) by a factor of 3 to account for the random orientation of the Plateau borders [12]. Since the only liquid being considered is that within the Plateau borders, the liquid fraction in the foam, ε, can be defined as the ratio of the volume of liquid in the Plateau borders and the volume of the system. Following [13], in order to convert the balance from being over a single Plateau border to being over a volume of foam containing many of these channels, the variable is defined as the local Plateau border length per unit volume, which results in ε = A.
(2)
The length of the Plateau borders per unit volume can be determined for foam cells of equivalent sphere radius R. The model in this paper makes use of a semi-empirical relation obtained from simulations of foams with random structure [14] that leads to =
3.3 R
4 3
R3
−1/3
4
+ 0.063
3
R3
−2/3
.
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Assuming that the density of the liquid is constant, the conservation of liquid can be expressed as
∂ε + ∇ · (εu) = 0. ∂t
(3)
The solution of Eqs. (1)–(3) provides a description of the liquid velocity, the Plateau borders cross-sectional area and the liquid fraction through the foam. The classic form of the standard foam drainage equation is easily obtained from substitution of the values for u and ε from Eqs. (1) and (2), respectively, into Eq. (3). In terms of boundary conditions, the bubbles at the liquid-foam interface can be considered to be nearly spherical and for randomly packed bubbles liquid fraction takes a value of approximately 0.36. For the walls and top surface the boundary conditions for liquid velocity can be expressed as no normal flow. In Fluidity, the Galerkin finite element method [15] is used to approximate the solution of the continuous foam drainage model by transforming the equations to a weak form, multiplying them by a test function and integrating over the domain. Rather than looking for a solution of the fields in the entire function space, a linear combination of basis functions is used. For the simulations in this work we use piecewise linear basis functions for the liquid fraction and the Plateau border cross-sectional area, and discontinuous piecewise constant basis functions for the liquid velocity field. A detailed description of the model and the reasons behind the selection of these particular basis functions for the fields in the foam drainage equation will be the subject of a separate publication. Our model also takes advantage of the anisotropic adaptive unstructured meshing techniques implemented in Fluidity [16,17]. The size and shape of the elements in the mesh can thus vary in order to dynamically resolve developing solution features. Adaptive mesh simulations of gravity current fronts, for example, have been shown to perform as well as high resolution fixed mesh simulations while using at least one order of magnitude fewer nodes and with a minimal cost for the mesh adapting [18]. Similarly, details of a performance analysis for simulations of liquid drainage in an upward flowing foam using Fluidity’s adaptive remeshing techniques can be found in [19]. The accurate tracking of the wetting fronts that develop in forced drainage experiments would also be greatly benefited from the use of anisotropic adaptive remeshing. Fig. 1 shows and example of a uniform forced drainage simulation that takes advantage of this technique.
Fig. 1. Simulation results for a uniform forced drainage experiment showing the liquid fraction (left) and the corresponding adapted mesh (right) at different simulation times. As the wave front propagates downwards, the size of the elements in the mesh is reduced in the vicinity of the wave front in order to resolve the dynamics.
3. Localised forced drainage simulations Contrary to standard uniform forced drainage, in which gravity effects are dominant, localised forced drainage experiments allow further investigation of the effect of capillary forces on the horizontal spreading of liquid in the foam. Localised forced drainage simulations were performed for a rectangular Hele-Shaw cell of width 12 cm and depth 0.3 cm. The domain is the same as the one studied in [4], although a wider range of widths for the liquid addition input, from 0.3 cm to 6 cm, were analysed in this work. A monodisperse foam was considered, with bubble radius R = 5 ×10−2 cm. The initial liquid fraction for all the simulations was set to 5 × 10−3 , and the liquid addition at the top of the foam was achieved by considering a source term for the liquid content. As in previous localised forced drainage experiments [4], the liquid addition flowrate per unit input width was kept the same as the width was varied. The flowrate of liquid addition for the smallest input width was 5 × 10−3 cm3 s−1 . For the simulations in this work the physical parameters were set as follows: = 1 ×103 kg m−3 , = 2.5 × 10−2 N m−1 and = 1 ×10−3 Pa s. The criteria employed for mesh adaptivity in these simulations is based on an interpolation error for liquid fraction, set to 5 × 10−4 . The edge length of the elements in the mesh was allowed to vary
with upper and lower limits of 5 × 10−1 cm and 5 × 10−3 cm, respectively. For further details on the mesh adaptivity procedure in Fluidity see [11,16]. Fig. 2 shows the simulation results for liquid fraction at time t = 10 s for input widths of 0.3, 0.6, 1.5, 3, 4.5 and 6 cm. The vertical position of the wave front was measured at the centre of the input addition zone. Results obtained by tracking the wave front for a liquid fraction ε = 0.02, can be seen for the different input widths in a double logarithmic plot in Fig. 3. The data adjust to a power law, with the behaviour of the front in the vertical direction being practically the same above an input width of 3 cm. For the horizontal spreading, rather than considering the position at a constant height as in previous experiments [4], the wave front was tracked at the position corresponding to the maximum horizontal value of the liquid fraction isocontour, which changes in height with time. Results for the horizontal position at different input widths are presented in Fig. 4. We found the data to be well described by a power law for t ≥ 4 s. The values of the exponent in the power law describing the position of the wave front in time are shown in Fig. 5 for the different input widths and for both the vertical and horizontal spreading. These values have been obtained from data at t ≥ 4 s. Since the
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. . . . . . . . Fig. 4. Horizontal position (from centre of the input width) of the wave front in the localised forced drainage simulations; the maximum horizontal value of the liquid fraction isocontour ε = 0.02 is tracked. Different symbols correspond to different input widths.
exponents are dependent on the isocontours of liquid fraction being tracked, we show the values for ε = 0.02, corresponding to the results in Figs. 3 and 4, as well as the values for isocontours of ε = 0.006 and ε = 0.05. The exponent for the vertical spreading increases with input width, in agreement with the experiments and simulations by Hutzler et al. [4]. This increase, however, is only marginal above an input width of 3 cm. On the other hand, as the width of the localised input increases the exponent in the power law characterisation of the horizontal position with time decreases. The data show a faster displacement in the vertical direction if the maximum isocontour value is tracked. For the horizontal position
Fig. 2. Liquid fraction numerical results at 10 s for input widths of: (a) 0.3 cm, (b) 0.6 cm, (c) 1.5 cm, (d) 3 cm, (e) 4.5 cm, and (f) 6 cm.
0.02
0.05
Power law exponent - vertical
0.006
. . . . . . . .
Fig. 3. Vertical position (from top) of the wave front in the localised forced drainage simulations; values are for an isocontour of liquid fraction ε = 0.02 and different symbols correspond to different input widths.
Power law exponent - horizontal
Input width (cm)
Input width (cm)
Fig. 5. Exponents in the power law fit for the horizontal and vertical position of the wetting front in the localised forced drainage experiments at different input widths.
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.
25
. . . . . . .
Fig. 6. Horizontal displacement (measured from the closest edge of the liquid addition zone) of the wave front in the localised forced drainage simulations; the maximum horizontal value of the liquid fraction isocontour ε = 0.02 is tracked. Different symbols correspond to different input widths.
the isocontour corresponding to the lowest liquid fraction value presents a faster displacement. The results for the horizontal position of the wetting front described above considered the distance of the front from the centre of the domain. If instead the actual displacement of the wetting front is tracked, i.e. that from the closest edge of the liquid addition zone, it becomes clearer that the data for the horizontal displacement of the front can be well described by a power law, as seen from the double logarithmic plot in Fig. 6. The exponent in the power law describing the horizontal displacement, as defined above, increases with input width only initially, with its value not varying significantly above an input width of 1.5 cm. This is shown in Fig. 7, in which values are presented for isocontours of different liquid fractions. Although the setup in these numerical experiments was similar to that presented in [4], a quantitative comparison of results would require information not reported in the aforementioned work (in particular the physical properties of the liquid in the simulations). However, we found qualitative agreement between the results for the vertical position of the front with regards to its power law behaviour.
Fig. 7. Exponents in the power law fit for the horizontal displacement of the wetting front in the localised forced drainage experiments at different input widths.
We have shown that the horizontal displacement of the wave front in these type of experiments is also well described by a power law when tracking the maximum horizontal value of the front and not that at a fixed height as in previous studies [4]. This is particularly clear if the actual displacement is analysed, and not the position with respect to the centre of the liquid addition zone. Our results also indicate an initial increase in the exponent characterising the power law for both vertical and horizontal displacement, but only up to a certain input width, above which the value of the exponent remains practically the same. The input width at which this is observed is, however, different for the vertical and horizontal displacements. 4. Uniform forced drainage simulations in non-rectangular containers Another variation of the classic forced drainage experiment arises if non-rectangular containers are considered. By changing the shape of the container different liquid fraction profiles can be obtained, which results in deviations from a uniform wetting front that moves with constant velocity. In this section we study the effect of four different container shapes on the dynamics of the wetting front when liquid is added all across the top of the foam.
Fig. 8. Non-rectangular container shapes considered for the forced drainage simulations: (a) converging planar walls; (b) diverging planar walls; (c) converging curved walls; (d) diverging curved walls.
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Fig. 9. Liquid fraction numerical results at 50 s, showing an isocontour of ε = 0.02, for containers with: (a) converging planar walls; (b) converging curved walls; (c) diverging planar walls; (d) diverging curved walls.
Vertical position (cm)
10 Converging curved-advanced front Converging planar-advanced front Converging curved-centre Converging planar-centre
1
0.1 1
10 Time (s)
100
Fig. 10. Positions of the wave front in the forced drainage simulations with converging walls; values are for an isocontour of liquid fraction ε = 0.02.
10
Vertical position (cm)
Simulations of uniform forced drainage experiments are carried out for Hele-Shaw type cells (5 cm in height and 0.3 cm in depth) with converging and diverging walls; for each case planar and curved walls are also evaluated. The geometries and dimensions of the four domains are shown in Fig. 8. The same physical properties for the liquid, bubble size, initial liquid fraction and addition flowrate per unit input width as in Section 3 were considered. Mesh adaptivity was also used in these simulations, with the same criteria employed for the localised forced drainage cases in the previous section. For clarity, the analysis of the results from the simulations are presented first for the containers with converging walls, followed by those with diverging walls. This is due to the similarities that the developing wetting fronts present. The liquid fraction profiles at 50 s for the four different containers are presented in Fig. 9, including an isocontour of ε = 0.02 for each case. It is observed that in containers with converging walls a high liquid fraction region develops at the walls, which travels down the foam at a higher velocity than the centre of the wave front. We refer to this high speed liquid jet as the advanced front. For containers with diverging walls, on the other hand, it is seen that the fastest region of the wave front is that at the centre of the container width, whilst at the walls a delayed front develops. Fig. 10 shows, in a double logarithmic graph, the vertical position of the wetting front for the containers with converging walls for t ≥ 4 s. The data can be well described by a power law, with different regions of the wetting front presenting differences in behaviour. The wetting front centre propagates at the same speed regardless of the presence of planar or curved walls. Curved converging walls result in an advanced front at the walls whose displacement is initially greater than the one for the converging planar walls, although it can be seen from Fig. 10 that the latter exhibits a higher power law exponent. The results for the position of the wetting front when using diverging walls are presented in Fig. 11 for t ≥ 4 s. As before, a power law behaviour was found. The displacement of the front at the centre, which is the fastest moving region for these configurations, is almost the same for both planar and curved walls. For the container geometries presented, the displacement of the delayed front in the case of diverging planar walls is greater than the one for diverging curved walls. Although this study considers only a few variations in container geometry, the finding that forced drainage in non-rectangular vessels can be well represented by a power law for the different regions
Diverging planar-centre Diverging curved-centre Diverging planar-delayed front Diverging curved-delayed front
1
0.1 1
10 Time (s)
100
Fig. 11. Positions of the wave front in the forced drainage simulations with diverging walls; values are for an isocontour of liquid fraction ε = 0.02.
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is encouraging for the carrying out of experimental and further numerical work. 5. Conclusions Simulations of localised forced drainage scenarios in rectangular containers have shown that the position of the wetting front with time is well described by a power law not only for the vertical spreading but also for the horizontal displacement due to capillary forces. The data also indicate that increasing the input width results in an initial increase in the exponent for the power law describing the displacement, which is followed by a plateau. The input width above which the value of the exponent stops changing significantly was found to be different for the vertical and horizontal displacements. Numerical investigations of uniform forced drainage experiments were carried out for non-rectangular containers. It was found that the position of the distinctive regions of the wetting front also obeys a power law behaviour. For containers with converging walls a high liquid fraction zone develops at the walls and moves faster than the centre of the wetting front. In the presence of diverging walls it is the centre of the front that moves faster, as the horizontal transport of liquid to lower liquid fraction regions due to the shape of the walls results in a delayed section of the wetting front. The displacement of the centre of the front was found not to be affected by the planar or curved nature of the walls for either converging or diverging wall configurations. The finite element implementation of the foam drainage equation used in this work makes use of adaptive remeshing techniques to capture the dynamics of the wetting front. Our model can be used to numerically study forced drainage for arbitrary container geometries. Future research directions include further analysis of the wetting front in localised forced drainage experiments, including the behaviour once the wetting front reaches the walls of the container, as well as exploring the effects of flowrate variations in non-standard forced drainage scenarios. Three-dimensional forced drainage simulations will also be performed in future work.
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