# Application of Population Balance Models in Particle-Stabilized Dispersions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Population Balance Equations

#### 1.2. Desorption Energy and Particle Coverage Degree

- (a)
- particle/drop collision efficiency, which depends on the film drainage process,
- (b)
- initial attachment efficiency, which considers the impact of the three-phase contact line formation,
- (c)
- particle attachment efficiency, which describes the ability of the particles to remain attached at the interface,
- (d)
- the drop coverage efficiency, which defines the system’s ability to prevent coalescence as a function of particle coverage.

## 2. Materials and Methods

#### 2.1. Experimental Investigations Stirred Tank

#### 2.2. Numerical Investigations

^{®}version 7.6a [40] was used for the solution, which solves the coupled mass balance with an adaptive discretization at each time step using the Galerkin-h-p algorithm. With the implemented models for drop breakage and coalescence processes in the software environment, the free model parameters (${c}_{1,b}$, ${c}_{2,b}$, ${c}_{1,c}$, and ${c}_{2,c}$) can be fitted to the experimental data with an implemented parameter estimation routine. Initial values for the four fitting parameters and an initial DSD must be specified. With a Dirichlet boundary condition, the value of the initial DSD for ${d}_{min}$ is defined: $f\left({d}_{min}\right)=0$. The numerical definition limit of the considered DSD was set to the minimum value of ${d}_{min}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m and the maximum value of ${d}_{max}=1000\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m. Analogous to the DDSD (Equation (7)), a Gaussian normal distribution with a mean of $\mathsf{\mu}=500\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m and a standard deviation of $\sigma =25\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m was chosen to represent the initial numerical DSD $f(d,t=0\phantom{\rule{0.166667em}{0ex}}\mathrm{s})$.

^{®}attempts to minimize the residual between experimental and simulated data using the relative root-mean-square deviation (RRMSD). In this work, the RRMSD was additionally used to indicate the deviations between the results of simulations and experiments.

## 3. Results and Discussion

#### 3.1. Experimental Results

#### 3.1.1. Impact of Nanoparticles on Physical Properties

#### 3.1.2. Experimental Drop Size Distributions

#### 3.2. Numerical Results

#### 3.2.1. Parameter Estimation

#### 3.2.2. Development of Modified Coalescence Efficiency Model

^{®}, it is possible to calculate the number of particles ${n}_{d}$ that theoretically must be removed at each time step of a simulation.

^{®}and via the case discrimination, ${c}_{np,c}$ is determined. With experimental data for $\Theta \ge 0.79$, the new fit parameter ${c}_{des}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}6.17\xb7{10}^{10}{J}^{-1}$ with an RRMSD of $\overline{r}=5.16\%$ of the implemented, modified coalescence efficiency equation was determined using the parameter estimation routine of Parsival

^{®}.

#### 3.3. Comparison of Simulated and Experimental Results

#### 3.3.1. Comparison of Simulated and Experimental Transient Sauter Mean Diameters

#### 3.3.2. Simulated and Experimental Sauter Mean Diameters and DSDs in Steady State

#### 3.3.3. Daughter Drop Size Distribution

## 4. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

C&T | work from Coulaloglou and Tavlarides [8] | |

DSD | drop size distribution | |

DDSD | daughter drop size distribution | |

PBE | population balance equation | |

RRMSD | relative root-mean-square deviation | |

TEM | transmission electron microscopy | |

w/o | water-in-oil | |

2D | two-dimensional | |

Symbols Used | ||

A | $\left[{\mathrm{m}}^{2}\right]$ | area |

a | [-] | constant in beta distribution |

$\dot{B}$ | $\left[{\mathrm{m}}^{-4}{\mathrm{s}}^{-1}\right]$ | birthrate in PBE |

b | [-] | constant in beta distribution |

C | [-] | measure of the width of the normal distribution |

c | [-] | fitparameter in PBE submodels |

${c}_{np,c}$ | [-] | dimensionless damping factor in coalescence efficiency |

${c}_{des}$ | $\left[{\mathrm{J}}^{-1}\right]$ | fitparameter in the energybarrier modell |

d | $\left[\mathrm{m}\right]$ | drop diameter (d${}^{\prime}$ and d${}^{\prime \prime}$ are used to differentiate between drops) |

${d}_{32}$ | $\left[\mathrm{m}\right]$ | Sauter mean diameter |

$\dot{D}$ | $\left[{\mathrm{m}}^{-4}{\mathrm{s}}^{-1}\right]$ | death rate in PBE |

D | $\left[\mathrm{m}\right]$ | tank diameter |

E | $\left[\mathrm{J}\right]$ | energy |

F | $\left[{\mathrm{m}}^{3}{\mathrm{s}}^{-1}\right]$ | coalescence rate |

f | $\left[{\mathrm{m}}^{-4}\right]$ | density function of number |

${f}_{d}$ | [-] | ratio of daughter drop volume to mother drop volume |

g | $\left[{\mathrm{s}}^{-1}\right]$ | breakage rate |

h | $\left[\mathrm{m}\right]$ | height |

H | $\left[\mathrm{m}\right]$ | height |

n | $\left[{\mathrm{s}}^{-1}\right]$ | stirring speed |

${n}_{p}$ | [-] | number of particles in the system |

${n}_{d}$ | [-] | desorbed number of particles |

P | $\left[\mathrm{J}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}\right]$ | power |

q | $\left[{\mathrm{m}}^{-1}\right]$ | density function |

Q | [-] | cumulative function |

r | $\left[\mathrm{m}\right]$ | radius |

s | $\left[\mathrm{m}\right]$ | strength |

t | $\left[\mathrm{s}\right]$ | time |

T | $\left[\mathrm{K}\right]$ | temperature |

V | $\left[{\mathrm{m}}^{3}\right]$ | volume |

w | [-] | mass fraction |

w | $\left[\mathrm{m}\right]$ | wide |

Greek Symbols | ||

$\beta $ | [-] | daughter drop size distribution |

$\eta $ | $\left[\mathrm{Pa}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}\right]$ | dynamic viscosity |

$\u03f5$ | $\left[\mathrm{W}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{kg}}^{-1}\right]$ | mass specific energy dissipation rate |

$\epsilon $ | [-] | voidage |

$\lambda $ | [-] | coalescence efficiency |

$\kappa $ | $\left[\mathrm{S}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}\right]$ | electrical conductivity |

$\nu $ | [-] | number of daughter drops |

$\phi $ | [-] | dispersed-phase volume fraction |

$\rho $ | $\left[\mathrm{kg}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}\right]$ | density |

$\gamma $ | $\left[\mathrm{N}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}\right]$ | interfacial tension |

$\dot{\gamma}$ | $\left[{\mathrm{s}}^{-1}\right]$ | shear rate |

$\sigma $ | $\left[{\mathrm{m}}^{3}\right]$ | standard deviation |

$\mathsf{\mu}$ | $\left[\mathrm{m}\right]$ | mean value |

$\theta $ | ${[}^{\circ}]$ | contact angle |

$\Theta $ | [-] | interface coverage degree by particles |

$\xi $ | $\left[{\mathrm{m}}^{3}{\mathrm{s}}^{-1}\right]$ | coalescence frequency |

$\zeta $ | $\left[\mathrm{V}\right]$ | zeta potential |

Subscripts | ||

$ads$ | adsorption | |

b | breakage | |

b | baffle | |

c | continous phase | |

c | coalescence | |

$cov$ | covered | |

d | dispersed phase | |

d | desorbed | |

d | daughter | |

$des$ | desorption | |

$eff$ | effective | |

$exp$ | experimental | |

$e/s$ | between endoscope and stirrer | |

g | gap | |

$in$ | initial | |

m | mother | |

$max$ | maximum | |

$min$ | minimum | |

$np$ | nanoparticle | |

o | organic phase | |

$o,w$ | oil/water interface | |

p | particle | |

$sb$ | stirrer blades | |

$sim$ | simulated | |

$st$ | stirrer | |

$stat$ | in steady-state | |

$tot$ | total | |

3 | volume-based |

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**Figure 1.**Normal (norm.), bimodal and beta DDSDs. Distributions are plotted with a variety of distribution width ($C=3$ and $C=5$) for bimodal and normal distributions.

**Figure 2.**Illustration of (

**a**) the planar interface ${A}_{p,cov}$ of a particle adsorbed at the liquid/liquid interface (purple marked), with a contact angle $\theta $, (

**b**) the void fraction $\epsilon $ of a hexagonal two-dimensional (2D) packed spherical particle layer (red marked).

**Figure 3.**Experimental setup, baffled reactor equipped with a Rushton turbine and in situ endoscope technique and enlarged view of the stirrer and the endoscope with the reflective Teflon device. The corresponding dimensions of the stirred tank set up including baffles (b) and stirrer blades (sb) are shown in the table.

**Figure 4.**Schematic visualization of the two-step parameter estimation procedure. Initial parameters (${c}_{1,b,in}$, ${c}_{2,b,in}$ for breakage, ${c}_{1,c,in}$ and ${c}_{2,c,in}$ for coalescence) were taken from another substance system. In the first step, the initial breakage parameters were fitted to experimental data where coalescence is completely halted (F = 0), resulting in two new breakage parameters ${c}_{1,b}$ and ${c}_{2,b}$. In a second step, these are kept constant, and the initial free parameters of coalescence were fitted to experimental data where coalescence and breakage occur (pure system), resulting in new coalescence parameters ${c}_{1,c}$ and ${c}_{2,c}$.

**Figure 5.**Suspension viscosities with decane as base fluid as a function of shear rate, with different particle mass fractions ${w}_{p}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.25$–$1\%$.

**Figure 6.**Experimental results: (

**a**) Transient Sauter mean diameter with a step-wise change in stirrer speed from $n=900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm to $n=700\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm at $t=1160$ s, (

**b**) stationary density-volume distributions ${q}_{3}$ at $n=900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm for particle mass fractions ${w}_{p}=0.25\%$, ${w}_{p}=0.75\%$ and ${w}_{p}=1\%$.

**Figure 7.**(

**a**) Transient calculated interface coverage degree for ${w}_{p}=0$–$1\%$ before and after an abrupt stirrer speed decrease at t = 1160 s, (

**b**) fitted parameter ${c}_{np,c}$ as a function of the coverage degree $\Theta $ by particles at $n=900\mathrm{rpm}$ (blue stars), fitted curve (black dotted line) ${f}_{empricial}$ for interface coverage degrees lower 0.9.

**Figure 8.**Number of particles ${n}_{d}$ that theoretically need to be removed from the interface during the coalescence process of two drops that are covered with particles. The energy needed to desorb particles from the interface is equal to an energy barrier against coalescence, based on [25].

**Figure 9.**Comparison of experimental and simulation results: Simulations (lines) with the use of the free parameters listed in Table 1 and modified coalescence efficiency (Equation (12)) and experimental data (symbols). (

**a**) Transient Sauter mean diameter with a step-wise change in stirrer speed from $n=900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm to $n=700\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm at $t=1160\mathrm{s}$, (

**b**) stationary density-volume distributions ${q}_{3}$ at $n=900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm for particle mass fractions ${w}_{p}=0.25\%,{w}_{p}=0.75\%$ and ${w}_{p}=1\%$.

**Figure 10.**(

**a**) Comparison between experimental (circles) and simulated (crosses) Sauter mean diameter for different stirrer speeds ($n=700,800,900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm) in steady state ($t=900\phantom{\rule{0.166667em}{0ex}}$ s), (

**b**) simulations (lines) in comparison with experimental data (symbols) of the volume based density function ${q}_{3}$ in steady state ($t=900$ s) for ${w}_{p}=1\%$ with a variation of stirrer speed. Simulations were performed with free parameters summarised in Table 1 and modified coalescence efficiency according to (Equation (12)).

**Figure 11.**Impact of DDSD shape (normal, bimodal and beta distributed) and width ($C=3$ and $C=5$) on volume-based density DSD ${q}_{3}$ for the case of binary breakage ($\nu =2$). Simulations were performed with the free parameters listed in Table 1 and with the modified coalescence efficiency (Equation (12)) for ${w}_{p}=1\%$ and $n=900\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$rpm in comparison with the experimental distribution. Experimental and simulation results are shown in steady state ($t=900$ s).

**Table 1.**Initial and fitted parameters for the C&T PBE submodels, using the two-step parameter estimation routine illustrated in Figure 4.

${\mathit{c}}_{1,\mathit{b}}$ | ${\mathit{c}}_{2,\mathit{b}}$ | ${\mathit{c}}_{1,\mathit{c}}$ | ${\mathit{c}}_{2,\mathit{c}}$ | |
---|---|---|---|---|

initial values | $6.012\xb7{10}^{-2}$ | $2.833\xb7{10}^{-2}$ | $1.060\xb7{10}^{-4}$ | $1.435\xb7{10}^{11}$ |

fitted values | $2.639\xb7{10}^{-2}$ | $1.218\xb7{10}^{-1}$ | $3.394\xb7{10}^{-1}$ | $4.245\xb7{10}^{13}$ |

**Table 2.**Relative root-mean-square deviations in percent between experimental and simulated steady-state Sauter mean diameters, by variation of stirrer speed n and particle mass fraction ${w}_{p}$. For each stirrer speed, the total RRMSD for all 5 particle mass fractions is also tabled.

Particle Mass Fraction ${\mathit{w}}_{\mathit{p}}$ [%] | 0 | 0.25 | 0.5 | 0.75 | 1 | Total |
---|---|---|---|---|---|---|

RRMSD (${d}_{32,stat}$) [%] n = 900 rpm | 2.24 | 2.31 | 2.41 | 2.96 | 0.03 | 2.23 |

RRMSD (${d}_{32,stat}$) [%] n = 800 rpm | 4.70 | 5.37 | 5.49 | 1.68 | 4.61 | 4.59 |

RRMSD (${d}_{32,stat}$) [%] n = 700 rpm | 5.14 | 11.05 | 7.29 | 2.28 | 15.02 | 9.30 |

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## Share and Cite

**MDPI and ACS Style**

Röhl, S.; Hohl, L.; Stock, S.; Zhan, M.; Kopf, T.; von Klitzing, R.; Kraume, M.
Application of Population Balance Models in Particle-Stabilized Dispersions. *Nanomaterials* **2023**, *13*, 698.
https://doi.org/10.3390/nano13040698

**AMA Style**

Röhl S, Hohl L, Stock S, Zhan M, Kopf T, von Klitzing R, Kraume M.
Application of Population Balance Models in Particle-Stabilized Dispersions. *Nanomaterials*. 2023; 13(4):698.
https://doi.org/10.3390/nano13040698

**Chicago/Turabian Style**

Röhl, Susanne, Lena Hohl, Sebastian Stock, Manlin Zhan, Tobias Kopf, Regine von Klitzing, and Matthias Kraume.
2023. "Application of Population Balance Models in Particle-Stabilized Dispersions" *Nanomaterials* 13, no. 4: 698.
https://doi.org/10.3390/nano13040698