Simulations with Dispersive Mixtures Model

Transcription

1 Simulations with Dispersive Mixtures Model P.J.A. Janssen, P.D. Anderson & G.W.M. Peters MT03.05 May 12, Introduction The combination of two different materials in a blend can yield interesting results. The food industry, the polymer processing industry and more generally, all mixing processes are in some way connected to the phenomena occurring. If the volume fraction of one of the two components is relatively small, one can distinct droplets in a matrix fase. The behavior of these droplets can be quite complex and include phenomena like stretching of droplets, breakup and coalescence. Varies models to describe droplet deformation have been suggested and further there is the coupling of the single droplet behavior to bulk parameters like the stress in the total dispersion. 2 Modelling of drop behavior The model used here is the one as suggested by Peters [5]. This is a phenomenological model, which describes the morphology changes of the dispersed fase during flow and couples the state of the droplets to the bulk stress. The major drop states are shown in figure 1. From top to bottom one can see in figure 1 respectively stretching, filament break-up, necking and coalescence. When stretching, droplets can be stretched into long, slender filaments. These filaments in turn can breakup into many smaller droplets. Droplets can also break in two, in a proces called necking. Finally two droplets can coalesce into one new, bigger droplet. 2.1 Parameters To avoid any confusing, the first thing to do is to make a summary of the most important parameters used in the model. The parameters are extracted from the description of the deformation of a droplet. R stands for the original radius of 1

2 Figure 1: Droplet deformation modes the droplet, that is in un-deformed state, while L and B stand for the greatest and smallest length parameters of the now stretched droplet. Furthermore there are the zero shear viscosities η m and η d of matrix and the droplet fase and the interfacial tension α between them. N denotes the number of droplets per unit volume, while φ stands for volume fraction of the dispersed fase. When the mixture is under flow, γ equals the local shear rate. The capillary number Ca is used as a dimensionless number to characterize the flow. Simplified can be said that is represents the ratio of the viscous stresses to the surface tension. The ratio between the viscosities is denoted by p. The stretch factor β is used to indicate the stretch of the droplet in comparison with its initial, equilibrium diameter. A spherical drop thus yields a β of 1 and stretched droplets have a β larger than 1. The last parameter used is the interfacial area Q between the dispersed and the matrix fase. This is a measure for the degree to which the drops are dispersed in the matrix. 2.2 Droplet deformation A droplet can deform in many ways. These include stretching, necking, breakup and coalescence. Below are given the mathematical formulations for these different 2

3 Table 1: Physical parameters used to characterize a drop phase dispersed in a continuous matrix. Parameter Definition φ Volume fraction occupied by the droplets N Number of drops per unit volume η d Zero shear viscosity of droplet phase η m Zero shear viscosity of continuous matrix α Surface tensions between droplet and matrix γ Local shear rate Ca = η m γ Capillary Number α/r p = η d η m Viscosity ratio β = L Stretch ratio 2R Q = 2π N B Rβ Interfacial surface area modes as included in the model used Stretching If a droplet deforms perfectly with an imposed velocity field, the deformation is termed affine. The stretch ratio of a droplet for affine deformation can be written as: d ln(β) dt = D i j m i m j. (1) The orientation vector if the droplet, m j, can be calculated from the eigenvector of the maximum eigenvalue of the rate of deformation tensor D j j. The deformation of droplets is, however not always affine. For non-affine deformation of a droplet in an elongational flow, a relation was derived by Stegeman et al [8]. This results van be simplified to: d ln(β) dt = f (β)d i j m i m j. (2) The current stretch ratio thus directly influences the rate of deformation of the stretch factor. Stegeman s result is used in this study for general flows, not just elangational. This is done by calculating the elongational component of the general flow substituted in Stegeman s formula. 3

4 2.2.2 Necking Now that a method is provided to describe the drop deformation, the next step will be to determine the deformations which will break the droplets up into smaller ones. Recalling the definition of the capillary number as the ratio of the deforming forces to the restoring forces, or the strength of the flow, one can define a critical capillary number, which represents a flow that is strong enough to break or neck an initial spherical drop into two droplets. The time required for an initial spherically drop to neck into two drop was determined by Grace [4] by studying two dimensional flows: ( ) R0 η m t neck 85.3 p (3) α Peters determined, by assuming a linear necking rate for capillary numbers between Ca crit and 2Ca crit, the rate of change of the equivalent spherical radius of a drop undergoing necking: d R dt = γ R p 0.45 Ca crit. (4) Additionally it is assumed that necking drops have a stretch ratio of β = Filament breakup For stronger flows, Ca > 2Ca crit, it is necessary to consider how drops that have been stretched into long filaments break into many smaller drops. If this happens, the elongational processes occur more quickly than the breakup processes. Janssen [6] found that when the cross sectional radius of a filament, B, dropped below a critical value determined by capillary wave instability growth, the filament would breakup into many droplets: B crit 0.04ω ( ) ηm ɛ 0.9 m p 0.45 (5) α If (B < B crit ) then: ( ) 3π 1/3 R avg R crit. (6) 2 After a breakup occurs, the drops are assumed to have a stretch ratio of β =

5 If a stretched droplet is placed into a static or stationary fluid, the breakup conditions are slightly different. For a static filament, Janssen [6] found: ( ) 3λm (p) 1/3 R avg R. (7) 4R Since such drops are breaking up under static conditions, the resulting drops are assumed to have a stretch ratio of β = 1. Experiments show the appearance of so called satellite droplets. These are smaller droplets emerging between the greater ones. In this analysis these are neglected Coalescence The final item included in the model is coalescence. Based on the work of Chesters [2], Peters [5] derived a relationship describing the rate of change of the equivalent spherical radius of a drop undergoing coalescence. When the local capillary number is below the critical capillary number, the drop radius changes according to: d R dt = exp ( ) 3RpCa 3/2 4h crit 4 γ φ R 3π. (8) After coalescence, the drop stretch ratio can be determined from the work of Cox [3] as: 5 (19p + 16) g (p, Ca) = (19p 4 (p + 1) ) 2 + ( ). (9) 20 2 Ca β = ( ) 1 + g (p, Ca) 2/3. (10) 1 g (p, Ca) 2.3 Constitutive model Now that all modes for a single droplet have been covered, one must link the behavior of a single droplet to the bulk stress. This can be done by modelling all droplets or by modelling one, average droplet, which represents the mean properties of all droplets. The choice has been made for the latter approach, but this introduces problems, like step wise behavior of certain parameters, like the stretch ratio and the interfacial surface, especially after breakup and coalescence and isn t very realistic, since droplets vary in size. To overcome these problems an effective measure parameter has been introduced. It links the average drop modelled 5

6 to a more realistic and smoother value for the mean of all droplets, using a simple exponential decay relation: τ Q Q ef f + Q ef f = Q; (11) τ β β ef f + β ef f = β. (12) The value of the time scale parameters associated with this effective measure approach have to be estimated from experiments. Furthermore these values vary for different states in which the drops/ filaments are, but these aren t very large, i.e. the order of magnitude is the same. To describe the stress for a dispersion of droplets in matrix, Peters [5] derived the following relationship: Tij = Pδij + [ 2 + ] 10 (η d η m ) φ η m D ij + S ij. (13) 2 (1 φ) η d + (3 + 2φ) η m Essentially, the volume average bulk stress, T ij is equal to an isotropic pressure term, P, a rate of deformation term, D ij which is modified by an effective Newtonian viscosity and an extra stress due to the presence of the drops, S ij. The extra stress is governed by the following differential equation where the first term, S (1) ij with: represents the lower convective derivative of the extra stress: ] ( S ij = L ijkl Qef f, φ, S ) D kl, (14) [ 1 S (1) ij + τ D Dt ln ( ) Q ef f L ijkl = 19µ d + 16µ c Q φ µ c 4 2 I + [5(1 φ)µ d + (5 2φ)µ c ] Q 0 τ 0 3 IS ij 2 Eσ Q S ij S ij.(15) In this equation σ represents the interfacial tension and E is a function of the zeroshear viscosities and the volume fraction. Equation(14) couples the extra stress to changes in the interfacial surface area, Q e f f. Additionally, the relaxation time τ, appearing in equation(14) is also a function of interfacial surface area and drop stretch ratio, β e f f as [ (βef 3φ f 1 ) ] { (19ηd + 16η m ) [2 (1 φ) η d + (3 + 2φ) η m ] τ = 4αQ ef f 10 (1 φ) η d + (10 4φ) η m With methods to model drop dynamics (i.e. deformation, necking, breakup and coalescence) and a constitutive model to connect drop morphology to bulk stress, specific applications can be investigated to test the abilities of this work to model drop dispersion behavior. 6 }.(16)

7 Table 2: Physical parameters for a PID/PDMS dispersion studied by Vinckier [9]. Parameter Value matrix component poly-dimethylsiloxane (PDMS) η m 195 Pa s τ m sec. droplet component poly-isobutene (PIB) η d 86 Pa s τ d sec. dispersion parameters φ 0.10 (PIB) α N/m initial morphology R m N m Computational methods The equations stated above were implemented and solved in the finite element package SEPRAN using a discrete elastic-viscous stress splitting, discontinuous Galerkin (DEVSS/DG) approach with first-order time integration. See the Bogaerds [1] for more detailed information. Summarized one can say that this is a computational method that solves all unknown variables concerning the macroscopic behavior simultaneously. In other words: the deformation and the stresses are coupled. Each two-dimensional element consisted of nine nodes with secondorder polynomials used as basis functions. For all of the simulations, physical parameters for the dispersion were taken from Vinckier [9] and are reproduced in table 2. 7

8 3 Step Shear In the step shear study, a 128 element, 561 node mesh was used with a time step of 0.01 seconds. Following the results of Schiek [7], some more simulations have been done in a step shear flow. Since step shear is a relatively simple flow, not only numerical, but results can also be easily interpreted, it is very suited for analysis of the influence of parameters. Before variations are investigated, the original results are depicted in figure 2 with explanation of the different zones. As one can see, the Number of drops per unit volume [m 3 ] Affine Stretching Necking Filament Breakup Necking balanced with Coalescence Time [s] Figure 2: Original result for step shear initial droplets are stretched until the value of B drops below the critical one and the drops breakup, resulting in a great increase in the number of droplets. After the breakup, necking is the most important morphology state. This again leads to an increase of droplets, but this time slower and more gradually. Finally the necking is balanced with coalescence, which leads to no netto increase in the number of droplets. 8

9 3.1 Shear rate First the influence of different values of the end shear rate is examined. Original a step was made from a shear rate of 0.3 s 1 to 3.0 s 1. The end value of the shear rate was made 30 s 1, 15 s 1 and 1.5 s 1. Initial morphology was as equilibrium in a shear rate of 0.3 s 1. In figure 3 the evolution of the number of droplets in time is depicted. As can be seen, a higher shear rate will lead to more droplets. This Number of drops per unit volume [m 3 ] Shear rate = 30.0 s 1 Shear rate = 15.0 s 1 Shear rate = 3.0 s 1 Shear rate = 1.5 s Time [s] Figure 3: The number of droplets in time for different shear rates is to be expected, since a higher shear rate leads to a higher shear stress and can more easily overcome the interfacial tension of the droplets. A shear rate of 30 s 1 and 15 s 1 leads to an intermediate platform in the number of drop after the first breakup. Apparently the shear rate is still high enough to continue to stretch the droplets until a second breakup occurs, after which the the droplets are so small that the interfacial tension will prevent them from stretching too far and breaking up again. In fact equilibrium between necking and coalescence is found. This intermediate platform is notably smaller in time then the stretching zone prior to the initial breakup. This has part to do with the modelling; the stretch factor after breakup is set to 1.5. So droplets have initially a higher stretch factor then in the original configuration, in which the stretch factor was 1. Furthermore the droplets 9

10 will be smaller, and thus the critical cross section of the stretched filament as stated in equation 6, which is independent of the drop radius, will be reached earlier, and thus the filament will break up earlier. After the breakup a necking regime is seen for a shear rate of 30 s 1, 15 s 1 and 3 s 1. As can be derived from equation 4 a higher shear rate will lead to a faster change in radius and thus faster breakup and more droplets. This zone is followed Number of drops per unit volume [m 3 ] Shear rate [s 1 ] Figure 4: The end value for the number of droplets for different shear rates by a area in which necking and coalescence are in equilibrium. If the number of drops in the end is plotted against the shear rate, a straight line is observed, as seen in figure 4 This is to be expected, as results in the past have clearly pointed out: Vinckier [9]. In here the radius of the droplets is plotted against the shear rate. The radius of the drops is reversibly proportional to the number of drops. This behavior was inserted in the model, so this is actually nothing more then a validation of the model. 3.2 Effective measure parameter In the model the step behavior of certain parameters were smoothed. The introduction of these effective measures lead to a time constant τ for the effective stretch 10

11 factor β and the interfacial area Q. The values of these constants were obtained from experiments, but under different conditions. The assumption that the value would be the same for a different situation doesn t hold automatically. To investigate if these assumptions still hold, the time constants were varied slightly. The objective is not to make the best fit, but rather study the influence Interfacial Area In the first series the parameter for the effective interfacial area Q is investigated. The results are shown in figure 5 together with the original. Besides the values for the effective interfacial area, the only variables plotted here are the normal and the shear stresses, in respectively figure 6 and figure 7, since they can be measured and are directly influenced by the effective measures. 3 x Interfacial Area [m 2 ] τ Q = 4 τ Q = 2 Original τ Q = 0.5 Unaffected value for area Time [s] Figure 5: Evolution of the effective value of the interfacial area for different time constants In the filament stretching zone there is no difference between the different values of τ. This is to be expected, since that no step wise changes in the interfacial area have occurred. The effective value will follow the real value. After the breakup 11

12 τ Q = 4 τ Q = 2 Original τ Q = 0.5 Normal Stress Time [s] Figure 6: Normal stress with different time constants for the effective value for the interfacial area things become more interesting. Increasing the value for τ Q, which means a slower decay to the real value, leads to a lower peak in the normal stress. This can be explained by the step the interfacial area makes towards a greater value, due to the smaller droplets, which have more area per volume than big droplets. A slower responding effective interfacial area will lead to a smaller value for the effective interfacial area. When the coalescence starts, the greater value for the time constant will again lead to a slow response. According to equation( 14) the interfacial area has a major influence in the relation describing the stress. If the effective value of the interfacial area doesn t undergo major changes, it is likely that the stresses also have a more gradual course. Of course the interfacial area is not the only parameter determining the stress, but a greater value for the time constant for the interfacial area will lead to a smoother course of the stress. It can also be noticed that the normal stress is influenced far more drastically than the shear stress. This is rather difficult to explain, but the 4th order tensor 4 L, as stated in equation( 15), has a part in it with Q on the diagonal in 4th order 4 I, which may dominate the normal stress, rather then the shear stress in the non-main diagonal parts of the stress tensor. 12

13 Shear Stress τ Q = 4 τ Q = 2 Original τ Q = Time [s] Figure 7: Shear stress with different time constants for the effective value for the interfacial area 13

14 3.2.2 Stretch factor Besides the time constants for the interfacial area, the time constants for the effective stretch ratio are varied also. The results are shown in figure 8. Besides the stretch ratio, the only variables plotted are the normal stress and the shear stress τ λ = 4 τ λ =2 Original τ λ =0.5 Real value for stretch ratio Stretch ratio Time [s] Figure 8: Evolution of effective stretch ratio with different time constants If the time constant for the effective stretch ratio β is varied, one can again see a slower responding stress for higher values for the time constant. The explanation for this is found in the value for the stress relaxation time as stated in equation( 16). A greater value for β leads to a higher value for the relaxation time. After the breakup the drop stretch ratio is set to 1.5. The effective measure parameter will decay from a value of approximately 30 just before the breakup to this end value. If the decay is even more slowed, then the value for the relaxation time of the stress will also be higher. A slower response of the stress is expected again and seen in figure 9 and figure 10. It is also seen that the relaxation has a major influence on both the normal stress and the shear stress. The stress relaxation time affects both. But as seen in equation( 16) the effective value for the interfacial value is also included in the stress relaxation time. Due to the power law coefficient of 1.5, which leads to a super-linear behavior for the effective stretch ratio, in comparison 14

15 Normal Stress τ λ = 4 τ λ =2 Original τ λ = Time [s] Figure 9: Normal stress with different time constants for the effective value for the stretch ratio to a linear behavior, in the numerator that is, for the effective interfacial area, will the stress relaxation time be influenced more by the stretch ratio then the interfacial area. Furthermore,the stretch ratio will have to make a greater decay, i.e. from 30 to 1.5, in comparison with the interfacial area, which varies within and m 2. All in all will parameters that affect β have a greater influence on the stress relaxation time then that parameters that affect the interfacial area do, and thus will have a greater influence on the general course of the stress. If one would like to make new fits for the time constants, a good approach would be to first fit the effective stretch ratio on the shear stress and then the interfacial area on the normal stress, followed by fine tuning. 3.3 End value of β As stated before, the model is not complete. For example: complex mechanisms like end-pinching and tip-streaming are not included. This has partly to do with the absence of a decent description of the phenomena, but nevertheless one can at least make an attempt. To multiply the end value of β with a constant, a very 15

16 Shear Stress τ λ = 4 τ λ =2 Original τ λ = Time [s] Figure 10: Shear stress with different time constants for the effective value for the stretch ratio crude first step has been made to model phenomena influencing the stretch ratio of droplets, a parameter with great influence. Once again it is not the objective to make the best fit, but to study influence. One of the most important tests for the model is the end value of the stress. Therefore the multiplication parameter has only been implemented in the coalescence modelling, since this is the most important phenomena occurring after the breakup, and thus will be of major influence on the end value for the stresses. The value of β, i.e the right hand side of equation 10, has been multiplied with 2 and 0.5. In figure 11 the results are shown for the normal stress. The shear stress results are seen in figure 12. As seen the normal stress is affected far more then the shear stress. One should be careful including these factors. 16

17 Normal Stress [Pa] Original β=2x org 50 β=0.5x org Data Vinckier Time [s] Figure 11: The influence of a multiplication factor for β on the normal stress Shear Stress [Pa] Original β=2x org β=0.5x org Data Vinckier Time [s] Figure 12: The influence of a multiplication factor for β on the shear stress 17

18 y x 4 Contraction Figure 13: Mesh for 4-to-1 contraction MESH scaley: scalex: An first approach has been taken to run the model in a 4-to-1 contraction. In advance one can say that there are a lot of problems to be expected. The combination of a corner singularity and a time-dependent problem is usually very problematic for numerical routines. The DEVSS/DG method used can handle singularities, but has stability problems with time dependent problems. But since Schiek [7] has managed to run his step shear and journal bearing flow, it is to be expected that the contraction will also work, although there were also problems reported. Furthermore there is a combination of shear flows and elongational flows. Some parameters are flow type dependent, especially the ones concerning critically capillary number. There are made no adjustments for this. 4.1 Mesh Initially, the mesh had a straight corner, but after some divergence problems a small corner radius of 0.1 was introduced in the mesh. The mesh included further 711 elements with 4898 free degrees of freedom. A picture of the mesh is seen in figure 13. The height from the symmetry line to the top wall is 1.6 and the the total length is 5. A parabolic velocity profile, with a maximum fluid velocity of 1.5 m/s at the center line, was prescribed at the inlet on the left side of the mesh. The top walls all had impenetrable, no-slip boundary conditions, i.e. the x- and y-velocity were set to zero. At the outlet the y-velocity was set to zero, as in a fully developed outflow. Simulations without this boundary condition justified this assumption. The bottom side of the mesh had the y-velocity set to zero, as in a symmetric flow. 18

19 LEVELS Contour levels of Stress 11 (TIME sec) Figure 14: Normal stress scaley: scalex: time t: Results Due to numerical problems it was not possible to complete a full simulation and reach an equilibrium state. The results represented below are after 0.5 seconds, with a maximum fluid velocity of 1.5 m/s in the symmetry line. Although this is definitely not equilibrium, it represents the general trends. The numbers for the state of the droplets mean: 0: Static filament 100: Filament stretching 200: Start filament stretching 300: Filament breakup 400: Static filament breakup 500: Coalescence 600: Coalescence after static breakup 700: Necking As one can see, the greatest values for the shear stress and for the stretch ratio are found in the contraction at the top wall of the outlet. This has lead to the greatest stretch ratios and hence at these zones the greatest concentration of droplets is expected. The major problems were identified, but no origin or satisfactory solution were found. The major problem was the value of the stretch factor dropping below 1, which gives for example problems in equation 16, were the power of 1.5 applied 19

20 LEVELS Contour levels of Shear Stress?? (TIME sec) Figure 15: Shear stress scaley: scalex: time t: LEVELS Contour levels of DMM Beta eff. (TIME sec) Figure 16: Effective stretch ratio scaley: scalex: time t: LEVELS Contour levels of DMM DPFLAG (TIME sec) Figure 17: State of the droplets scaley: scalex: time t:

21 to a negative number will lead to problems. In most of these points the trace of the rate of deformation tensor D was not zero anymore, which is in violation of the incompressibility constraint. Artificial solutions, like building in constraints to prevent the stretch factor dropping below 1, have been inserted. This lead to a slightly longer simulated time, but there was an ever increasing number of points in which the stretch factor dropped below 1, resulting eventually in divergence of stress to infinity. There have been made attempts by refining the mesh to run for longer time, but these effects were not satisfactory enough. Mesh refinement did lead to slightly longer simulated times, but the computational time was out of proportion and thus there have been made no attempts to refine the mesh to extreme. Reducing the time step also didn t lead to major improvements. The best results were obtained with a reduction of the mean fluid velocity with a factor of ten, in combination with a reduced time step and some mesh refining. The simulation ran longer, and more importantly the total fluid flux, velocity times total simulated time, was greater by about a factor 2. However since the velocity was smaller, and thus local shear rates were smaller, there were no phenomena observed that didn t show in the original. 21

22 5 Conclusions and Recommendations The dispersive mixtures model as suggested by Peters [5] shows promising results. The implementation in a step shear simulation has been successful and results have proven to be realistic. A higher shear rate leads to more and smaller droplets. Adjusting parameters for the effective measures for the interfacial area and stretch factor can greatly influence the stress and thus need to be chosen with care. Simple adjustments to the stretch ratio also influence the stress. All these parameters adjustments should be handled with care. An attempt to run the model in a 4-to-1 contraction has proven to be unsuccessful. Numerical problems lead to divergence of stress and no satisfactory solution has been found. But trends found indicate generally in the right direction, of many small droplets found along the walls. To run the contraction problem successful, one could decouple the deformation, the stresses and the drop morphology. For this, the used DEVSS/DG method would have to be replaced with a decoupled routine. Furthermore the deformation of the droplets should be study in a more controlled way, with lots of checks during the simulation. Since the stretch ratio tends to drop below 1 at some points, there are probably faults in the calculation of the deformation of the droplets. So a thorough check of these can perhaps yield some progress. References [1] A.C.B. Bogaerds, 3D Viscoelastic Analysis, Masters Thesis, Department of Mechanical Engineering, Eindhoven University of Technology, [2] A.K. Chesters, The modeling of coalescence processes in fluid-liquid dispersions: A review of current understanding Transactions of the Institute of Chemical Engineers, 69A pp , [3] R.G. Cox, The deformation of a drop in a general time-dependent fluid flow Journal of Fluid Mechanics, 37 pp , [4] H.P. Grace, Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. 3rd Engineering Foundation Conference on Mixing, Republished in Chemical Engineering Communications, 14 pp , [5] G.W.M. Peters, S. Hansen & H.E.H. Meijer, Constitutive modeling of dispersive mixtures Journal of Rheology, 37(3) pp ,

23 [6] J. Janssen & H.E.H. Meijer, Droplet break-up mechanisms: Stepwise equilibrium versus transient dispersion Journal of Rheology, , [7] R.Schiek, P.D. Anderson & G.W.M. Peters, Simulating Polymer Blends with a Phenomenologically Derived Constitutive Model, Internal Report, Section Materials Technology, Department of Mechanical Engineering, Eindhoven University of Technology, [8] Y. Stegeman, F.N. van de Vosse, A.K. Chesters & H.E.H. Meijer, Break-up of (non-) Newtonian droplets in a time-dependent elongational flow Proceedings of the Polymer Processing Society s Hertogenbosch, The Netherlands (CDrom), 15, [9] I. Vinckier, Microstructural Analysis and Rheology of Immiscible Polymer Blends, Ph.D thesis, Katholieke Universiteit Leuven, Faculteit Toegepaste Wetenschappen, Departement Chemische Ingenieurstechnieken, de Croylaan 46, 3001 Leuven, Belgium,