Drop Size Distributions as a Function of Dispersed Phase Viscosity: Experiments and Modeling

2.1 Impact of Dispersed Phase Viscosity on Drop Size Distributions

Rising dispersed phase viscosity at otherwise constant conditions shifts the DSD towards larger diameters while the Sauter mean diameter d_3,2¹⁾ and maximum diameter d_max increase ⁵. A higher drop viscosity enhances resistance against deformation which affects the energy needed for breakage. This results in a reduced breakage rate at constant energy input ¹³. The dispersed phase viscosity determines the inner movement of the drop, which in turn affects the drag coefficient, the relative velocity of the droplet and the velocity boundary condition of the drainage flow. Additionally, the internal flow of the drop can influence the film drainage ⁶. The reduction of the breakage rate is more pronounced than the impact on the coalescence rate, and the overall drop sizes increase with rising dispersed phase viscosity.

In contrast to d_max, the minimum diameter d_min decreases, which leads to a changing shape of the overall drop size distribution, e.g., from a normal to a lognormal shape ⁵. This effect is caused by the rising elongation of the mother drop resulting in a higher number of satellite drops and a higher probability of unequal-sized daughter drops ⁵, ⁶. In general, one breakage event can lead to two or more daughter drops, depending on the substance system and the specified period for one breakage event.

According to Zhou et al. ⁶, who investigated dispersed phase viscosities up to η_d = 315 mPa s, binary breakage still is dominant with 60–80 % of breakage events in systems with high dispersed phase viscosity. Due to these different effects, accurately describing the balance between breakage and coalescence phenomena as a function of dispersed phase viscosity is a challenging task that can be tackled using PBEs.

2.2 Population Balance Models for the Description of Dispersed Phase Viscosity

The change of a drop size distribution over time can be calculated using a balance of birth and death terms ^1-4 for drops of a certain size formed or destroyed due to breakage and coalescence phenomena. The general PBE structure for a liquid-liquid system in a discontinuously operated stirred tank without mass transfer or reaction is used in this work. For information concerning the general PBE framework, refer to ¹, ⁴, ¹².

Effects that need to be expressed by submodels are the coalescence rate described via the coalescence efficiency λ and the drop-drop collision frequency ξ, the drop breakage rate g, the daughter drop size distribution β, and the number of daughter drops per breakage event ν. Using the assumption of ideal mixing, local dependency of the drop size distribution can be neglected. Depending on the chosen submodel, these terms contain different physical properties and process parameters ², ³, ¹⁴.

The most widely used submodels are the ones by Coulalouglou and Tavlarides ¹. However, they do not consider the dispersed phase viscosity, and the interface is treated as immobile. For the substance systems investigated in this work, the assumption of a partly mobile interface is more realistic ¹⁵. Hence, a literature search was conducted for available models that consider partly mobile interfaces and include dispersed phase viscosity as a physical property in coalescence and breakage rate.

Submodels for breakage rate are provided by Zhou et al. ⁷, Podgórska and Bałdyga ¹¹, and Alopaeus et al. ¹², ¹⁶. Submodels for coalescence efficiency for partially mobile interfaces that account for dispersed phase viscosity are provided by Chesters ¹⁰, Podgórska and Bałdyga ¹¹, and Alopaeus et al. ¹², ¹⁶. In this work, only the submodels by Alopaeus et al. ¹², ¹⁶ are employed to describe breakage and coalescence. Still, a brief description of the other models, including their advantages, is presented in the Supporting Information.

The breakage rate submodel by Alopaeus et al. ¹⁶ is a further development of the statistical model by Narsimhan et al. ¹⁹, assuming breakup due to collisions and critical velocity fluctuations around the drop surface ², ¹⁸. In addition, a dependency on turbulent dissipation rate was added to the eddy/drop collision frequency, taking into account the viscous force of fluid inside the drop, which has been neglected in most previous work. The resulting drop breakage rate g by Alopaeus et al. ¹⁶ includes three fit parameters c_1,b, c_2,b, and c_3,b:

(1)

The collision frequency ξ (Eq. 2) determines how often droplets collide with each other and includes an additional fit parameter c_1,c:

(2)

The coalescence efficiency λ is defined according to the film drainage model and is based on a model by Tsouris and Tavlarides ³, ²⁰ for partial-mobile interfaces. The critical film thickness where rupture occurs is considered negligible compared to film thickness at the start of the drainage process. The initial film thickness is directly proportional to the harmonic mean of the two colliding drops. The coalescence efficiency is described as:

(3)

with

(4)

using the physical properties of the continuous phase (η_c, ρ_c) and dispersed phase (η_d, ρ_d), a fit parameter c_2,c and a modified energy dissipation rate ε_mod defined as

(5)

The parameters c_1,b, c_2,b, c_3,b, c_1,c, and c_2,c of the PBE can be fitted to experimental data. This use of several fit parameters in PBE models is very widespread and needed to counteract inadequacies of the physical description, e.g., due to applied simplified assumptions or due to the still limited understanding of all details of breakage and coalescence processes which can lead to deviations between experiments and simulations. In this work, the PBE submodels by Alopaeus et al. ¹², ¹⁶ will be evaluated in detail in combination with a discussion of the fit procedure and an analysis of the impact of daughter drop size distribution (DDSD) shapes on simulation results.

2.3 Daughter Drop Size Distribution

The DDSD function can be described using statistical, phenomenological, or empirical models ¹, ⁷, ²⁰, ²¹ and directly affects the PBE simulation results, especially the simulated DSD shape. A representative selection of DDSD shapes is presented in the following, but many other options are available in the literature. Coulaloglou and Tavlarides ¹ used a Gaussian DDSD with a mean value

(6)

expressed by the mother drop volume V_m divided by the number of daughter drops per breakage event using ν = 2 for binary breakage. The distribution function is:

(7)

using the dimensionless daughter drop volume

(8)

and the standard deviation σ

(9)

with an adjustable parameter c describing distribution width. Another option is to use a beta distribution and adjust its parameters to form a broad bell shape ²¹. This can be achieved by using an expression with a gamma function:

(10)

including variable parameters p and q ¹⁶, ¹⁷, ²¹. The broad bell shape reduces the probability of equal-size daughter drops compared to the Gaussian distribution, which is often considered more realistic. Based on single-drop breakage experiments in a water-toluene system, Maaß et al. ²¹ proposed a bimodal (M-shaped) DDSD due to the assumption that binary breakage into two equal drops is not energy-efficient. The width of the two distribution peaks can again be varied with the parameter c implemented in the standard deviation σ.

(11)

Tsouris and Talvarides ²⁰ proposed a phenomenological model in which this trend towards breakage into unequal daughter drop sizes is further pursued. For equal-sized breakage, a maximum additional interfacial energy ΔE_max is necessary resulting in a probability density function minimum. The highest probability density is obtained for the most unequal daughter drop sizes, resulting in a U-shaped distribution. For details, refer to ²⁰. The dimensionless DDSD is given by:

(12)

A recent empirical model was developed by Zhou et al. ⁷, ⁹ using experiments in a pump-mixer to investigate the breakage process and the number of daughter drops as a function of different parameters including dispersed phase viscosity. Multiple breakage events were thereby treated as a series of binary breakups, which were divided into three different types: original tensile breakup, intermediate tensile breakup, and revolving breakup. The equations for each type of breakup are provided in ⁷. A modified Weber number We_mod was defined as:

(13)

with

(14)

to determine what type of breakage is dominant at specific experimental conditions. All three types of breakage are described as a weighted sum and We_mod is used to weigh the different terms. Fig. 1 compares all DDSD types and the specific parameters used for this visualization, and all subsequent simulations. The impact of DDSD on the simulated Sauter mean diameters and DSD shapes and resulting deviations from experimental data will be discussed in Sect. 4.3.

3.1 Stirred Tank, Endoscope Technique, and Image Analysis

A double-walled stirred tank equipped with four rectangular baffles and a Rushton turbine was used for all experiments (Fig. 2). The temperature was controlled using an external thermostat and geometrical parameters were kept constant. Water and the respective type of oil were filled into the tank before the agitation started. Oil-in-water emulsions were achieved with dispersed phase volume fractions φ = 0.05 and 0.15. Experiments were performed using different agitation speeds (mean energy dissipation rates, respectively). The stirrer speeds were n = 500 rpm (ε = 0.320 m²s⁻³), n = 600 rpm (ε = 0.554 m²s⁻³), and n = 700 rpm (ε = 0.879 m²s⁻³).

Energy dissipation rates were determined via torque measurements. Impeller Reynolds numbers Re_st were calculated using the mixture density and viscosity considering the respective dispersed phase fraction. The resulting values ranged from Re_st = 520–6540, indicating a transitional flow regime. Due to the small tank volume, the dispersion can still be considered homogeneous throughout the tank.

An endoscope measurement technique and subsequent automated image analysis (SOPAT GmbH) were used to determine the drop size distributions of the agitated systems in situ. The endoscope was connected to a camera (GX 2750, Allied Visions Technology). The outer diameter of the endoscope was 0.021 m. To improve image quality, a diffuse PTFE reflection adapter was attached to the endoscope tip with a gap size of 0.006 m. The endoscope tip was positioned with a horizontal distance of 0.01 m to the stirrer blade tips to ensure optimal flow of liquids through the gap. For details concerning the endoscope probe and the impact of different reflection adapters on image quality, refer to ²² and ²³. The measurement range of the endoscope was 25–1000 µm. Every drop size distribution (or corresponding Sauter mean diameter) shown in the results was calculated using at least 5000 measured drop diameters from a series of 200 images.

A sensitivity analysis was performed to ensure that enough drops were detected for statistical certainty. During this analysis, the change of characteristic diameters of the experimental DSD was determined as a function of the number of detected drops. Fluctuations of these values decrease with rising drop number until only extreme values such as the d_max might still be affected by further image analysis. This sensitivity towards largest diameters is often visible as tailing of experimental DSD but is of little statistical relevance. Using a frame rate of 11 s⁻¹, the acquisition of one image series took approximately 18 s. The measurement error is 7–8 % of the Sauter mean diameter. To maintain graph clarity, error bars will be only shown in selected graphs of the results section but are also representative for other process conditions.

3.2 Numerical Solver for PBEs

The commercial software Parsival (Computing in Technology GmbH) ²⁴, ²⁵ was used to solve the PBEs. It uses the Galerkin h-p method based on a generalized finite-element scheme with self-adaptive grid- and order construction. Conservation laws, which ensure that the total volume of the dispersed phase (i.e., the total volume of all drops) is maintained during simulation, still must be satisfied after the discretization with respect to the internal coordinate. The intelligent adaptive algorithm used by Parsival controls the discretization of the internal coordinate depending on the shape of the distribution. The conservation of mass and volume is ensured using several control loops and heuristics already implemented in the software. For more details concerning refer to ^24-26.

PBE model fit parameters (c_1,b, c_2,b, c_3,b, c_1,c, and c_2,c) can be fitted to experimental data with an implemented parameter estimation routine while minimizing the residual (relative root-mean-square deviation, RRMSD) between experimental and simulated data. Ideally, the estimation converges nicely with a low RRMSD and results in a unique solution. The considered drop size range for all simulations was between d_min = 1 µm and d_max = 1000 µm (based on experimental experience). A Gaussian normal distribution with a mean of 500 µm and a standard deviation of 25 µm was used as the initial drop size distribution. It was ensured that the daughter drop sizes were within the aforementioned drop size range and that the daughter drop sizes were always smaller than the respective mother droplets. Simulations were performed with a single zone PBE using mean energy input values for the whole vessel (see Sect. 3.1).

4.1 Exemplary Experimental Results

Not only steady-state Sauter mean diameters and DSDs were determined in experiments, but also their reaction to dynamic conditions induced by an abrupt variation of stirrer speed. The advantage of dynamic experiments is that additional information about breakage and coalescence rates can be gathered. The combination of steady state and dynamic experiments is also exploited in the PBE fitting procedure described in Sect. 4.2.

Fig. 3a shows Sauter mean diameters of the different silicone oil-water systems in dynamic experiments. Agitation is started at t = 0 s with n = 700 rpm and switched to n = 500 rpm at t = 1800 s. Silicone oil M50 (lowest dynamic viscosity) has the smallest Sauter mean diameters and quickly reacts to changes in agitation speed by running into a new steady state. The balance between breakage and coalescence phenomena is shifted towards coalescence with rising dispersed phase viscosity and it takes longer to reach a new steady state. For M350 and M500, the drop diameters at n = 500 rpm became too large, preventing a reliable image analysis. Therefore, data for these conditions are not shown.

Fig. 3b depicts the volume-based density distributions for all silicone oil-water systems within the first steady state (n = 700 rpm, t = 1800 s). Rising drop sizes with higher dispersed phase viscosity are also clearly visible here. The heights of distribution peaks reduce while distributions are shifted towards larger diameters. Tailing of experimental DSD at large diameters can be caused by single large drops, e.g., in case of the last data points at large diameters of the M200 distribution (see also Sect. 3.1). The results follow expectations and literature ⁶, ⁷. Further experimental results will be shown and discussed in combination with the simulation results in the following sections after the PBE fitting procedure is described.

4.2 Fitting Procedure and Resulting Fit Parameters of the PBE Model

The fit of PBE parameters for breakage (c_1,b, c_2,b, c_3,b) and coalescence (c_1,c, c_2,c) to experiments was performed as an iterative process. The aim was to use only a small amount of experimental data for the fit procedure and to define one fixed set of fit parameters for all oil-water systems. A fit to only one experiment did not lead to satisfactory simulation accuracy of all systems. Therefore, a second experiment was included in the fit procedure based on the idea that a fit to one rather breakage-dominated experiment and to one rather coalescence-dominated experiment might lead to better simulation results (Fig. 4).

The breakage-dominated experiment was M50 oil, n = 700 rpm, φ = 0.05. The coalescence-dominated experiment was M200 oil, n = 500 rpm, φ = 0.05. The parameters were iteratively fitted to the Sauter mean diameters over time and to the steady-state DSD at the respective stirrer speed. The fitting procedure started by fitting breakage parameters to the M50 experiments and coalescence parameters to the M200 Sauter mean diameters. Subsequently, the fit was optimized using the steady state DSD for M50 and M200. A Gaussian DDSD was used since it is most frequently applied in literature and the one initially suggested for the general PBE framework, but its impact will be further discussed in Sect. 4.3. A suitable set of fit parameters was identified (Tab. 2).

The fit result in comparison to fit experiments is illustrated in Fig. 4. It should be noted that a perfect description of dynamic and steady-state conditions for both experiments is not possible even with directly fitted parameters, indicating that the physical description of the PBE model is imperfect. For example, the fitted Sauter mean diameters for M200 oil are slightly lower than the experimental values. Still, the simulated steady-state DSD does not accurately account for the impact of higher oil viscosity on distribution shape, since d_max is too tiny in simulations. This is partly compensated by a higher peak at intermediate drop sizes. Since the fit parameters are somewhat interdependent, it is also possible that another set of fit parameters exists which would represent the experimental results even better. The following discusses the impact of rising dispersed phase viscosity, dispersed phase fraction and stirrer speed in all oil-water systems is predicted with these fit parameters.

4.3 Simulation Results

Since the description of the steady-state DSD shape for higher viscosity oils is problematic already in the fit procedure, the impact of DDSD on the simulation results and the possibility of achieving a better agreement with experiments are discussed. Subsequently, the model performance in dynamic experiments is evaluated.

Fig. 5 compares experimental and simulation results for the M50 and M500 oils (lowest and highest dynamic viscosity, respectively) as a function of DDSD type. Note that the combination of M50 at this process conditions with the Gaussian DDSD was used in the fitting procedure, but the other cases were not. For systems with M50 oils (Fig. 5a), the Gaussian DDSD leads to the most minor deviation between experimental and simulated data, followed by Beta DDSD and Zhou DDSD. The Tsouris DDSD leads to a different type of distribution with a slightly indicated second peak. This effect is even more pronounced if the bimodal DDSD is used but does not correctly describe the experimental monomodal distribution.

In the M500 system (Fig. 5b), the experimental distribution is shifted towards larger diameters while the distribution peak is lower, and a trend towards a bimodal shape is visible. Gauss and Beta DDSDs predict a very high and narrow peak for the M500 system, while the bimodal distribution significantly overestimates the second peak's height. Therefore, the DDSDs by Zhou and Tsouris are most suitable to describe the experimental distribution in this case. Although the Zhou DDSD was specially designed to account for the dispersed phase viscosity, in the case presented here, the DDSD by Tsouris showed a slightly better agreement with experimental data. This may be caused by differences in the experimental setup since the DDSD by Zhou was developed for a pump-mixer ⁷, ⁹. It should also be noted that both DDSD types predict a broader DSD than the experimental one for low viscosity and a narrower DSD for high viscosity, indicating that the shift of d_min and d_max with rising dispersed phase viscosity (see Sect. 2.1) is not simulated correctly. Using individual fit parameter sets for each DDSD type should further improve simulation accuracy and hence might change the conclusion that the Tsouris DDSD is the best option. This aspect might be tackled in future work.

A further comparison, including the oils with intermediate viscosity, is illustrated in Fig. 6. The experimental and simulated DSD for M50, M200, and M500 (Fig. 6a) indicate that the simulation with Tsouris DDSD represents the experimental data better than the Gaussian DDSD also for intermediate viscosity. Furthermore, the comparison of steady-state Sauter mean diameters for all oil types at different stirrer speeds and dispersed phase fractions shown in Fig. 6b illustrates that simulations using the Tsouris DDSD are the most precise. The fit parameters were determined using a Gaussian DDSD, but all other DDSD types in combination with these fit parameters showed better simulation results for higher viscosity.

Independent of the DDSD type, the highest deviations in the parity plot occur if the dispersed phase fraction is changed from φ = 0.05 to 0.15. An example where this is clearly visible is provided in Fig. 7 for M50 oil with Gauss and Tsouris DDSD. The combination of model and DDSDs (or the used fit parameters) does not sufficiently emphasize the impact of dispersed phase fraction. As expected, an increase in dispersed phase fraction leads to higher Sauter mean diameters and shifts the DSD towards larger diameters while the peak height decreases. The same trend is also visible in simulations, but a higher dispersed fraction's impact is clearly underestimated. This effect is even more pronounced for higher viscosity oils (data not shown), indicating that the new balance between coalescence and breakage rate due to turbulence damping and higher drop/drop collision frequencies ^27-29 with rising φ is not adequately shifted towards coalescence.

Fig. 8 shows a comparison of experiments and simulation both in dynamic experiments and in steady state for all oils at a dispersed phase fraction of φ = 0.05 using the DDSD by Tsouris. The simulation of Sauter mean diameters (Fig. 8a) works quite well for low-viscosity oils (M50, M100, M200).

The time needed to reach a steady state is slightly overestimated for M50 and M100, whereas the results are in better agreement for high-viscosity oils. However, deviations of the steady-state value occur, especially for M350 and M500 oils. For these cases, high simulated Sauter mean diameters indicate that the balance between coalescence and breakage rate is shifted too far towards coalescence, which was not the case for variation of φ. Interestingly, the Sauter mean diameters are overestimated for M50 but underestimated for M100 and M200 after the change in agitation speed. Probably the description of the energy dissipation rate in the model could be subject to optimization, e.g., by using common approaches such as compartment modeling ³⁰ or by coupling the PBE with CFD ³¹. Nevertheless, the steady-state DSDs for 700 rpm (Fig. 8b) are represented quite well for all oil-water systems.

Fig. 9 shows exemplary results for the dynamic change of the DSD in M200 systems. In the first step, four data points were chosen from the dynamic experiments at n = 700 rpm (t = 0, 60, 180, 1800 s, see Fig. 9, top). Note that due to the limited frame rate in experiments, a specific point in time specifies the beginning of the time interval where the respective drop images series was taken. Since a simulation at t = 0 s would only result in the predetermined initial DSD (see Sect. 3.2), the simulation depicted for t = 0 s was actually done at t = 9 s (median value of the corresponding experimental time interval).

In accordance with the Sauter mean diameter values, experimental DSDs are shifted towards smaller drop diameters with time while the height of the peak increases. The simulations for the first two points in time already show a deviation from the initial Gaussian distribution shape towards the slightly bimodal shape governed by the Tsouris DDSD. However, these two simulations do not represent the experimental data well due to the high peaks at large drop sizes. This can either be caused by the lasting effect of initial DSD or because the simulation underestimates the breakage rate in relation to the coalescence rate. However, Sauter mean diameters of the experiment and simulation are in good agreement. The simulation accuracy of the DSD improves for simulation times above t = 60 s, and experimental data are represented adequately, especially considering the higher experimental fluctuations towards the end of the experiment visible in Sauter mean diameter values.