5.5 Emulsion rheology
5.6.3 Simulation without settling experiment
Since the Henschke model necessitates either the initial experimental sedimentation velocity or the initial exper-imental Sauter mean diameter during sedimentation to calculate the coalescence curves, a prediction without an experiment is not possible unless these parameters are already known or can be estimated with sufficient accuracy. To estimate the difference between the steady state Sauter mean diameter and the Sauter mean diameter during sedimentation, the droplet sizes in agitated tanks were measured in steady state and in the sec-onds after agitation stop, as illustrated in Figure 70 for aC4E2system. The agitation stop measurements were repeated until enough droplets were analyzed in each time interval to reach a statistical relevance. However,
0 0.02 0.04 0.06 0.08 0.1 0.12
0 50 100 150
Height h [m]
Time t [s]
0 0.05 0.1 0.15 0.2
0 50 100 150 200
Height h [m]
Time t [s]
Δtsep≈ 27 s fit. r*v,min
exp. 2
fit r*v,max(d3,2(t=0) = 372 μm) exp. 7 (d3,2(t=0) = 71 μm) a)
b) Δtsep≈ 30 s
Fig. 69: Settling curves and fit for experiment 2 and 7 (cf. Tab. 21)
0 50 100 150 200 250
0 1 2 3 4 5
Sauter mean diameter d3,2[μm]
time after agitation stop [s]
increase by a factor of≈4
Fig. 70: Increase of Sauter mean diameter after agitation stop at t = 0 s in 2
̄Φ condition (water, 1-dodecene,C4E2,α
= 0.5,γ= 0.2, T = 40◦C, initial agitation speed n = 700 rpm).
due to the influence of the endoscope gap size on the drop sizes under these inhomogeneous conditions, these results should be treated only as an indicator for the order of magnitude. In the first seconds after agitation stop numerous drop/drop coalescence phenomena occur and the Sauter mean diameter increases by a factor of approximately four, which is in proximity to the aforementioned deviations between the experimental steady state and calculated Sauter mean diameter during sedimentation.
The deviation between experimental steady state and Sauter mean diameter during sedimentation can be expressed as
d3,2(t=0)=d3,2(ss)+Q (45)
by introducing a summand Q that accounts for the diameter increase in the first seconds after agitation stop.
The deviations between steady state Sauter mean diameter and the calculated Sauter mean diameters during sedimentation are depicted in Figure 71 for the twelve experiments investigated before. According to Figure 71 the deviation decreases with rising temperature and increases with energy dissipation rateϵ. Therefore, the following expression was used to describe Q:
Q=q1
1
T +q2ϵ. (46)
To determineq1andq2 for this specific solvent system, the deviation between simulated and experimental data is minimized using the maximum error S3.
S3=max|(tE,sim,n,norm−1)| (47)
-200 -100 0 100 200 300 400 500
1 2 3 4 5 6 7 8 9 10 11 12
Sauter meandiameter d3,2[μm]
Experiment number [-]
V = 1.2L V = 2.5L
T↑
V = 0.3L
n↑
T↑ sim. 500 rpm sim. 700 rpm sim. 900 rpm exp. 500 rpm exp. 700 rpm exp. 900 rpm
n↑
n↑
Fig. 71: Experimental Sauter mean diameters derived from steady state measurements and calculated ones from the Henschke model (cf. Tab. 21)
0.25 0.5 0.75 1 1.25 1.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Norm. separation time tsep,norm[-]
Experiment number [-]
V = 1.2L V = 2.5L
T↑
n↑
α= 0.3 α= 0.5
T↑ T↑
n↑ experiment
Henschke + Q Henschke
Fig. 72: Experimental separation times and calculated results using the normal Henschke model and the Henschke model plus modifikation with Q for different experiments in water, 1-dodecene,C4E2 systems
This was performed for all experiments in Table 21 and leads to a maximum error of 23% in separation time.
If q1 and q2 are known, the calculation of the sedimentation velocity vs can be performed with the droplet swarm model using the experimental steady state Sauter diameters modified by Q. The separation time is then calculated with the known parameterr∗v,n. Figure 72 shows that the results obtained with the Henschke model and Q are in agreement with previous experimental results and additional data with other agitation speeds.
Experimental effort can be reduced since the pure Henschke model necessitates an experimental sedimentation velocity for every simulation. The simulation with Q allows to use the steady state Sauter diameters of agitated vessels to calculate the separation curves while the error is only slightly higher than the 20.2% achieved with r∗v,n. The values derived with Q have a tendency towards an overestimation of the separation times which could be optimized in further studies by adapting the definition of Q.
This also offers the opportunity to estimate the sedimentation velocity in systems where the endoscope technique can not be applied. An example for this is shown in Figure 73 where experimental separation times for an o/w C4E2 system are depicted by data points. The separation times were determined in the small V = 0.3 L vessel after stirring at n = 800 rpm. The energy dissipation rate was estimated with Equation 13 using Ne = 5 for a Rushton turbine, the stirrer diameter dst = 0.031 m and the volume and stirrer speed of the vessel. The Sauter mean diameters were calculated with the PBE model in Parsival using the fit parameter set for C4E2 systems (cf. Tab. 12) and then used to calculate the separation times. The results with the Henschke model with r∗v,weighted including Q are represented by the gray data points. The prediction leads to errors below 20% (∆tsep = 13 s) in comparison to the experimental data, which is a good result considering the simplified assumptions and possible error propagation. The sedimentation times of the complete droplet swarm in these
10 100 1000
20 30 40 50 60 70 80
Time t [s]
Temperature T [°C]
exp swarm
swarm (4x Sauter) Henschke + Q (PBE)
α= 0.3 γ= 0.2
Fig. 73: Experimental separation time of the complete system, calculated separation time with Henschke model + Q and swarm sedimentation based ond3,2 and 4xd3,2 (water, 1-dodecene,C4E2,α=0.3,γ = 0.2, V = 0.3L, DN60)
systems were calculated for comparison using the Sauter mean diameters and the height of the aqueous phase after separation as an approximation for the possible distance that a rising organic droplet might need to cover.
The swarm sedimentation times are larger than the experimental separation times of the complete system, which is caused by the lack of drop/drop coalescence in the calculation that would enhance the sedimentation speed.
Additionally the assumption of a specific sedimentation distance is not valid for all droplets of the swarm.
Furthermore, the swarm sedimentation time assuming that a four times larger initial Sauter mean diameter occurs due to drop/drop coalescence was calculated. In this case, the experimental sedimentation times are underestimated, which shows how sensitive the sedimentation is towards droplet size.