5.3 Drop size distributions in two phase systems
5.3.2 Two phase systems with surfactant
50 μm
Fig. 45: Microscope image of a water, 1-dodecene, Marlophen NP6 system at T = 20◦C,α= 0.5,γ= 0.05 (Zeiss Axio imager)
sizes, the image was taken at the edge of the emulsion sample where only one layer of droplets occurred. The image quality in case of several droplet layers is indicated in the top right corner of the image. Therefore, the in-situ experiments under 2
̄Φ conditions were mainly performed with the model systemC4E2. Figure 46 shows experimental and simulated Sauter mean diameters in a system with different compositions and agitation speeds in the 2
̄Φ temperature range of T = 20 - 78◦C [V]. All systems are o/w emulsions. The physical properties and simulation parameters are summarized in Table 11. Due to declining densities, viscosities and interfacial tensions the Sauer mean diameters also reduce over temperature. A normalized graph of the physical properties and Sauter mean diameter variation over temperature forα= 0.3,γ= 0.2 can be found in [III].
Figure 47 illustrates the standard deviation of the distributions normalized with the Sauter mean diameters.
Similar to the findings in [73] the values deviate around a value of σdsd/d3,2 ≈ 0.3 with a tendency towards higher values, which is typical for strongly coalescing systems. At these high surfactant concentrations, all interfaces are completely covered with surfactant molecules, since the aqueous phase has up to 20 - 40 wt.%
of surfactant (cf. section 5.1.3). Hence, partly uncovered interfaces are unlikely and the interfacial tension is only a function of temperature. The impact of rising energy dissipation rates is clearly visible in Figure 46, although the measurements forα= 0.5,γ = 0.4 show deviations at higher temperatures. The simulations indicated by the dashed line in Figure 46 were performed using the water/1-dodecene parameters (Fit 2). These simulations underpredict the Sauter mean diameters, which is a contrary result in comparison to the water/1-dodecene/Marlipal 24/70 systems where the experimental drop sizes were overpredicted. The reason for this behavior seems to be the low interface rigidity of the C4E2 systems which results in large droplet sizes in comparison to the Marlipal 24/70 systems due to higher coalescence rates. Even if the interfacial tensions in these systems are in the same range, the drop sizes can differ by an order of magnitude [IV]. Therefore, the model underpredicts the occurring coalescence in systems withC4E2. The solid lines show the modeling results with fit parameters directly from theC4E2 system (α= 0.3,γ= 0.2, n = 500 rpm, T = 30◦C) (Tab. 12).
Table 11: Simulation parameters of the systems as functions of T [◦C] valid for T = 20 - 78◦C. (aq = continuous phase, org = dispersed phase)
parameter α= 0.3,γ=0.2 α= 0.4,γ=0.2 α= 0.5,γ=0.4
σaq/org[N/m] -4.4315E-05 T+4.3537E-03
ρaq [kg/m3] (-6.0E-04 T + 1.015)*1000 (-6.0E-04 T + 1.0137)*1000 (-8.0E-04 T + 1.0045)*1000 ρorg[kg/m3] (-8.0E-04 T + 0.795)*1000 (-7.0E-04 T + 0.7843)*1000 (-8.0E-04 T + 0.7892)*1000
ηaq [Pa s] 0.0176 T−0.682 0.0312 T−0.859 0.488 T−1.35
ηorg[Pa s] 0.0149 T−0.716 0.0101 T−0.628 0.017 T−0.754
φ[-] 0.32 0.58 0.37
ϵ[m2/s3] ϵ500rpm= 0.341;ϵ700rpm= 936;ϵ900rpm= 1.989
Naturally, the simulation results are improved by a direct fit to the respective system and all parameter variations are adequately predicted by the model, except for the deviating Sauter mean diameters for high temperatures at α = 0.5,γ = 0.4. The comparison of the two sets of fit parameters (Table 12) shows a rising coalescence parameterc1,cas well as an increase in the breakage parameterc1,b, which mainly affect the dynamic behavior of the systems. In contrast,cc,b declines andc2,cincreases. The system-specificC4E2 parameters enhance both coalescence and breakage rate in comparison to the water/1-dodecene fit, while also affecting the ratio between
0 50 100 150
20 40 60 80
Sauter meandiameterd3,2[μm]
Temperature T [°C]
a) 2φ α= 0.3
γ= 0.2 used for
parameter fitting
0 50 100 150
20 40 60 80
Sauter meandiameterd3,2[μm]
Temperature T [°C]
2φ α= 0.4
γ= 0.2 b)
0 50 100 150
20 40 60 80
Sauter meandiameter d3,2[μm]
Temperature T [°C]
500 rpm
500 rpm (Sim C4E2 Fit) 500 rpm (Sim W/D Fit 2) 700 rpm
700 rpm (Sim C4E2 Fit) 700 rpm (Sim W/D Fit 2) 900 rpm
900 rpm (Sim C4E2 Fit) 900rpm (Sim W/D Fit 2)
c) 2φ α= 0.5
γ= 0.4
Fig. 46: Experimental and simulation results of water, 1-dodecene,C4E2systems for different compositions, temperatures and stirrer speeds using either the water, 1-dodecene parameters of Fit 2 or system-specific C4E2 parameters (derived fromα= 0.3,γ= 0.2, n = 500 rpm, T = 30◦C, V = 2.5 L, DN150) [V] (Parsival⃝R)
the two maximum values of g(dp) and F(dp). A higher coalescence rate in relation to the breakage rate is achieved, which results in the rising simulated Sauter mean diameters.
5.3.2.1 Analysis of different fitting procedures and comparison of the numerical PBE solution methods
All of the previous results already indicate that the choice of fit parameters is one of the crucial aspects of modeling with PBEs, especially because four different parameters are necessary. Up to this point, the time-dependent Sauter mean diameters were used for the fitting procedure and only Parsival⃝R was applied. Another option is to fit the empirical constants directly to the drop size distribution. The advantages and disadvantages of the respective fitting procedures were already discussed in section 3.3.4. The implementation of the PBEs in Matlab was mainly performed because of a simpler procedure of fitting c1,b,c2,b, c1,c and c2,c to the drop size distributions. It should be noted at his point, that the different numerical solution of the PBEs in Parsival⃝R and Matlab does lead to several deviations.
If the same process parameters are implemented in both softwares, the coalescence and breakage rates are the same but the resulting Sauter mean diameters differ. For the parameter set depicted in Table 13 the Sauter mean diameters were d3,2 = 97 µm in Parsival and d3,2 =122 µm in Matlab. These deviations are mainly caused by the simplified implementation of the daughter size distribution in MatLab which leads to a Gaussian distribution with a lower number of classes.
Figure 48 shows how the choice of experimental data affects the fitting parameters. For the fit parameters indi-cated by the data points, a normal distribution was chosen to represent the experimental drop size distributions and then used to fit the four empirical constants of the Coulaloglou and Tavlarides model in Matlab. The normal
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
40 60 80 100 120
σdsd/ d3,2[-]
Sauter mean diameter d3,2[μm]
500 rpm 700 rpm 900 rpm +15%
-15%
Fig. 47: Standard deviation normalized with the Sauter mean diameter for alle experimental Sauter mean diameters of Figure 46 (water, 1-dodecene,C4E2,α= 0.3 - 0.5,γ= 0.2 - 0.4, n = 500 - 900 rpm, T = 30 - 78◦C, V = 2.5 L, DN150)
Table 12: Fit parameters for 2Φ surfactant systems water, 1-dodecene C4E2 fit parameter (Fit 2)
c1,c 2.6918E+00 16.792
c2,c 1.3214E+14 7.354E+13
c1,b 9.0137E-03 0.081
c2,b 6.7809E-02 0.133
distribution is not the only option to fit the experimental curves, but was chosen here as a simplified approach.
The resulting ratio ofc1,b/c1,chas values between 1.5 and 6.19, whereasc2,branges from 2.30·10−3to 8.60·10−2 andc2,cfrom 1.8·1012 to 6.20·1013. The experimental results include different compositions, temperatures and stirrer speeds under 2
̄Φ conditions in C4E2 systems (Tab. 15). Simulating the Sauter mean diameters with every fit parameter value ofc2,b while leaving the other three parameters constant leads to deviations between d3,2 = 25 - 85µm. For all differentc2,c with constant other parameters the Sauter mean diameter ranges from d3,2 = 40 - 71µm. However, in most cases not only one empirical parameter changes during fitting. Therefore, these deviations neutralize themselves if several or all fitting parameters are affected by a fitting procedure. For example a widening or thinning of the distribution can lead to the same Sauter mean diameters. Due to the differences between the Matlab and Parsival⃝R calculations, the fitting procedure, drop size distributions and Sauter mean diameters cannot be directly compared. Therefore, a comparison is performed using the span of the resulting distributions where the diameter values of Q0 = 0.9 and 0.1 are divided by the arithmetic mean diameter of the distribution (Tab. 14). The width of the simulated distribution is too small if the fit parameters are derived from the time-dependent Sauter mean diameters. Differences between the span resulting from the water/dodecene Fit2 and the C4E2 fit occur, but since they cannot be controlled these values are quite ran-dom. The representation of the experimental distribution width is improved by the distribution fitting method.
However, deviations still occur since a normal distribution was used, which does not exactly represent the ex-perimental distribution type in all cases. These circumstances should be kept in mind, since the description of the actual distributions becomes especially important if the 3Φ conditions are reached, as will be described in the next subsections.
1E+12 1E+13 1E+14
0 5 10 15 20 25
Empiricalconstantc2,c[-]
Experiment number [-]
arithm. mean
0.001 0.01 0.1 1
0 5 10 15 20 25
Empiricalconstantc2,b[-]
Experiment number [-]
arithm. mean
1 10
0 5 10 15 20 25
Ratio of empiricalconstantsc1,b/c1,c[-]
Experiment number [-]
arithm. mean
α= 0.4 γ=0.2 α= 0.5 γ=0.2
α= 0.5 γ=0.4
α= 0.3 γ=0.2
Fig. 48: Fit parameters derived from a large set of experiments by fitting the experimental drop size distributions with a normal distribution in comparison (water, 1-dodecene,C4E2,α= 0.3 - 0.5, γ = 0.2 - 0.4, n = 500 - 900 rpm, T = 30 - 78◦C, V = 2.5 L, DN150)
Table 13: Parameters used for validation
c1,c c2,c c1,b c2,b ϵ[m2s−3] φ[-] σ[N/m] ρc[kgm−3] ρd[kgm−3] ηc[Pa s]
3.0E-03 1.3E+13 4.5E-03 4.6E-03 0.341 0.31 3.2E-03 997 771 1.76E-03
Table 14: Comparison of experimental and simulated width of the drop size distributions
Experiment W/D Fit 2 C4E2fit Matlab fit (mean) Matlab fit
fitting method - - - d3,2 = f(T) distribution
d90,0−d10,0
d1,0 0.815 0.367 0.228 0.678 0.53
Table 15: Overview of the experiments used for distribution fitting (V = 2.5L, DN150). Physical properties are summa-rized in Tab. 11
Nr. α γ T stirrer speed Nr. α γ T stirrer speed
[-] [-] [◦C] [rpm] [-] [-] [◦C] [rpm]
1 0.3 0.2 30 500 13 0.4 0.2 40 500
2 0.3 0.2 30 700 14 0.4 0.2 40 700
3 0.3 0.2 30 900 15 0.4 0.2 40 900
4 0.3 0.2 75 500 16 0.5 0.2 20 500
5 0.3 0.2 75 700 17 0.5 0.4 30 500
6 0.3 0.2 75 900 18 0.5 0.4 30 700
7 0.4 0.2 20 500 19 0.5 0.4 30 900
8 0.4 0.2 20 700 20 0.5 0.4 50 500
9 0.4 0.2 20 900 21 0.5 0.4 50 700
10 0.4 0.2 30 500 22 0.5 0.4 70 500
11 0.4 0.2 30 700 23 0.5 0.4 70 700
12 0.4 0.2 30 900