Stirred tanks for dispersion, the pre-dispersion of two immiscible liquids or particulate solidliquid suspension are extensively used in the chemical, food, pharmaceutical and metallurgical industries, for purposes such as suspension/emulsion polymerisation, heterogeneous/phase-transfer catalytic chemical reactions, paint production and hydrometallurgical solvent extraction. The aim of this paper is to propose the simple dispersion model enabling the prediction of particle size changes over time and taking into account the type of breakup mechanisms, the non-homogeneity of local turbulent energy dissipation rate in an agitated vessel and the effect of the number of times the liquid passes through the impeller and the impeller zone. The model was successfully tested on data published by Ditl et al. (1981)

Ditl, Pavel

Šulc, Radek

Portal Czasopism Naukowych (E-Journals)

 PAVEL DITL, RADEK ŠULC 
DISPERSION KINETICS MODELLING 
 
MODELOWANIE KINETYKI DYSPERGOWANIA 
 
 
 
A b s t r a c t   
Stirred tanks for dispersion, the pre-dispersion of two immiscible liquids or particulate solid-
liquid suspension are extensively used in the chemical, food, pharmaceutical and 
metallurgical industries, for purposes such as suspension/emulsion polymerisation, 
heterogeneous/phase-transfer catalytic chemical reactions, paint production and 
hydrometallurgical solvent extraction. The aim of this paper is to propose the simple 
dispersion model enabling the prediction of particle size changes over time and taking into 
account the type of breakup mechanisms, the non-homogeneity of local turbulent energy 
dissipation rate in an agitated vessel and the effect of the number of times the liquid passes 
through the impeller and the impeller zone. The model was successfully tested on data 
published by Ditl et al. (1981).  
Keywords:  dispersion kinetics modelling, solid particle-liquid dispersion, tooth impeller   
S t r e s z c z e n i e   
Mieszalniki do sporządzania mieszanin lub dyspergowania płynów niemieszających się są 
szeroko stosowane w przemyśle chemicznym, spożywczym, farmaceutycznym, 
metalurgicznym i wielu innych gałęziach przemysłu. Celem pracy jest przedstawienie 
prostego modelu dyspersji umożliwiający prognozowanie zmian wielkości cząstek w czasie z 
uwzględnieniem różnych mechanizmów rozpadu, niejednorodności lokalnych burzliwych 
szybkości rozpraszania energii w mieszalniku oraz liczby przejść płynu przez obszar 
mieszadła. Model ten został z powodzeniem przetestowany na danych opublikowanych przez 
Ditl et al. (1981). 
Słowa kluczowe:  modelowanie kinetyki dyspersji, układ dyspersyjny cząstki ciała stałego-
płyn, mieszadło ścinające 
DOI: 
                                                     
 Prof. PhD. DSc. Eng. Pavel Ditl, Assoc. prof. PhD. DSc. Eng. Radek Šulc, Department of Process 
Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague.  
84 
 
1.  Introduction  
Stirred tanks for dispersion, the pre-dispersion of two immiscible liquids or particulate 
solid-liquid suspension are extensively used in the chemical, food, pharmaceutical and 
metallurgical industries, for purposes such as suspension/emulsion polymerisation, 
heterogeneous/phase-transfer catalytic chemical reactions, paint production and hydrome-
tallurgical solvent extraction 1-3. Mixing plays a fundamental role in these systems 
controlling processes such as blending, homogenisation, mass transfer and chemical 
reactions etc. The quality of product, yield and economy of the processes is therefore 
significantly affected by mixing. Insufficient or excessive mixing may lead to wastage of 
processing time and raw material and/or the formation of by-products 4, 5.  
The prediction of mean drop/particle size and drop/particle size distribution (DSD) is 
vital for emulsification, suspension polymerisation, solid particle dispersion or 
crystallisation. Important process variables such as particle/drop size and size distribution 
are strongly affected by turbulence kinetic energy dissipation rate  and its distribution in 
an agitated vessel.  
2.  Theoretical background   
2.1. Liquid-liquid dispersion   
If acting mechanical forces are greater than drop interface surface forces, the drops 
deform and split into two or more smaller drops. Since this process is random, different 
sizes of daughter drops are produced. After a period of time, the equilibrium drop size 
distribution that is independent of the previous history is reached. Two possible 
mechanisms of dispersion are usually considered: drop break up by viscous forces and drop 
break up by inertial forces. The drop breakup mechanism is controlled by the turbulent 
eddy size 6. 
If the breakup occurs in the viscous subrange, i.e. dp  K, the drop breakup is 
controlled by viscous forces, hence the drop size dp can be expressed as follows 7: 
 
  21


cc
p
d , (1) 
where  
  – the local turbulent energy dissipation rate;  
 K – the Kolmogorov length microscale;  
 c, , c – the physical properties of continuous fluid,  i.e. density, interfacial ten-
sion, dynamic viscosity respectively. 
If the breakup occurs in the inertial subrange, i.e.    dp  K, drop breakup is 
controlled by inertial forces, hence the drop size dp can be expressed as follows 8: 
 
525353  
cp
d , (2) 
85 
 
To characterise equilibrium drop distribution, many authors used Sauter mean diameter 
d32. Assuming that   N
3·d2, Equation (2) is rewritten into a dimensionless form as 
follows: 
 
53
32
Wedd , (3) 
where  
 d – the impeller diameter;.  
 We – the Weber number; We = c·N
2·d3/.  
 
 
The mentioned correlations are only valid for dilute dispersions with a non-viscous 
dispersed phase where the viscous energy within the drops is negligible compared to its 
surface energy and the coalescence phenomenon is negligible. However, in the 
concentrated coalescing dispersions, the drop size is affected by both the breakup and the 
coalescence processes. It is assumed that the breakup occurs in the impeller region whilst 
the coalescence occurs mainly in the bulk region (e.g. 9). Moreover, the dispersed-phase 
viscous forces contribute to the drop stability when d  c.   
Many correlations describing the dependence of the Sauter diameter on the physical 
properties of fluids take both of the influences above mentioned into account by the relation 
in the following general form: 
       nWeViffdd 
2132
11 , (4) 
where  
  – the dispersed-phase volume fraction in the system;  
 Vi – the viscosity group, Vi = d/(d·d·)
1/2
;  
 n – the Weber number´s exponent.  
 
 In the breakup region the Weber number´s exponent is 0.6 8. In the coalescence region 
the exponent n is 3/8 10. Many authors use the Weber number´s exponent as an adjustable 
parameter. For example, Desnoyer et al. 11 take into account the effect of the dispersed-
phase volume fraction  on d32 considering the effect of the dispersed-phase fraction on the 
Weber number exponent n and on the constants of proportionality. According to Bałdyga 
and Podgórska 12, drop size depends on the fine-scale intermittency when the turbulence 
has intermittent character. In this case, the value of the Weber exponent depends on a 
intermittency scaling exponent.   
The published correlations for d32 assume steady state. The time to reach the 
equilibrium drop size distribution depends on the number of times the liquid passes through 
the impeller zone and the bulk zone. Following this fact the number of passages through 
these spaces is crucial for dispersion kinetics modelling as confirmed by Šulc and Ditl 13 
for flocculation kinetics modelling.     
Currently, the most common approach to predicting drop size distribution and its time 
evolution in agitated vessels is based on population balance models (e.g. 14, 15). 
86 
 
However, the quality of the prediction strongly depends on the quality of used breakage and 
coalescence kernels and the estimation of the local turbulent energy dissipation rate.   
2.2. Solid particle-liquid dispersion   
Compared with liquid-liquid dispersion, the mechanisms of solid particle-liquid 
dispersion is less complicated due to the rigid character of solid particles, especially when 
broken particles do not re-aggregate. In this case, only the breakup process occurs.  
The aim of this paper is to propose a simple model of dispersion kinetics for the 
prediction of the time course of solid particle size taking into account the type of breakup 
mechanisms, the non-homogeneity of the local turbulent energy dissipation rate in an 
agitated vessel and the effect of the number of times the liquid passes through the impeller. 
The model was successfully tested on data published by Ditl et al. 16.    
3.  Kinetics model    
For particle breaking occurring in a mechanically agitated vessel in the inertial 
subrange, i.e.   dp  K, the following assumptions were postulated: 1) the breaking 
occurs during passage through the impeller only; 2) broken particles do not re-aggregate; 
3) the minimum particle size is limited by the Kolmogorov length microscale K 
corresponding to the mixing intensity in the impeller zone; 4) the change of particle size 
can be described by first-order kinetics as follows: 
 )(
d
)d(
pfp
p
ddk
t
d
 , (5) 
where  
 k – the breaking rate-constant;  
 dp – actual particle size in time t;  
 dpf – the final particle size.  
 
Integrating Equation (5) for the initial condition dp (t = 0) = dp0 the time dependency of 
particle size dp can be obtained as follows:  
 )exp(
0
tk
dd
dd
pfp
pfp



, (6) 
where  
  dp0 – the initial particle size.  
 
Taking the number of passages of liquid through an impeller into account, Equation (6) can 
be rewritten as follows: 
87 
 
 )exp(
0
tNk
dd
dd
d *
pfp
pfp*
p



 , (7) 
where  
 k
*
 – the dimensionless breaking rate-constant; 
 dp
*
 – dimensionless particle size; 
 N – the impeller rotational speed.  
 
The Kolmogorov length microscale K is defined as follows:  
 
41
3
/
K 







 , (8) 
where  
  – the kinematic viscosity of agitated liquid; 
  – the turbulent energy dissipation rate.  
 
The turbulent energy dissipation rate in the impeller zone was estimated from impeller 
power input using swept volume approach and assuming that the practically whole impeller 
power is dissipated in the impeller zone. Thus, the final particle size was estimated as 
follows: 
 
41
3
41
3
)()(
)(
/
imp
/
imp
impKpf
V/(Pm/P
d


















 
 , (9) 
where  
 P – the impeller power input; 
 (P/m)imp – the specific turbulent energy dissipation rate in the impeller zone;  
 Vimp – the volume occupied by the rotating impeller. 
4.  Data analysis    
The proposed model was tested on data published by Ditl et al. 16. Ditl et al. 16 
studied the time course of the dispersion of TiO2 particles in liquid. The effect of impeller 
rotational speed, impeller type, vessel shape, impeller diameter to vessel diameter ratio, and 
solid particle concentration on dispersion was investigated. The effect of impeller rotational 
speed was investigated in a square vessel measuring 0.15 m × 0.15 m agitated by a tooth 
impeller having D/d = 4.54. The authors’ attention was also focused on the scale-up of 
dispersion. On the basis of results derived for the disc impeller by Schlichting 17, the 
authors proposed scale-up rule N·d0.89 = const. using which the turbulent shear strength on 
tooth impeller surface should be maintained constant. The scaling experiments were carried 
out in cylindrical vessels of various vessel diameters D agitated by a tooth impeller having 
D/d = 3.  
88 
 
The mentioned experimental results were used in this work for the testing of the 
proposed model. In accordance with the model assumptions, particle sizes measuring less 
than the estimated Kolmogorov length microscale were not taken into account. The 
evaluated model parameters are presented in Table 1. The comparison of experimental and 
predicted dependence of particle size on time is presented in Fig. 1 for data measured in the 
squared vessel and in Fig. 3 for data measured in the cylindrical vessel. The comparison of 
experimental data and the time course predicted by the model are presented in 
dimensionless form for the squared and cylindrical vessel in Fig. 2 and Fig. 5, respectively. 
The effect of dimensionless time N.t taking the number of times the liquid passes through 
the impeller into account is visible in Fig. 5. The curves of dp
*
 = f (N·t) are more compact 
than the dependencies dp
*
 = f (t) presented in Fig. 4.       
 
T a b l e  1  
Model parameters – data Ditl et al. 16   
 
  N  
[RPM] 
(P/m)imp 
[W/kg] 
k
*
 
[-] 
dpf  
[m] 
R
2
  
[-] 
K1 SQ150-N5000 5 000 4 714 0.764·10-5 3.816 0.9563 
K2 SQ150-N6500 6 500 10 337 1.476·10-5 3.136 0.9969 
K3 SQ150-N7500 7 500 15 867 1.574·10-5 2.818 0.9882 
K4 SQ150-N8800 8 800 25 613 4.392·10-5 2.500 0.9981 
SC1 D300-N3500 3 500 6 440 1.705·10-5 3.530 0.9955 
SC2 D190-N5550 5 550 10 309 1.549·10-5 3.138 0.9873 
SC3 D150-N7600 7 600 16 509 1.988·10-5 2.790 0.9556 
SC4 D100-N11500 11 500 25 485 1.514·10-5 2.503 0.9278 
 
 
89 
 
 
Fig. 1. Experimental data Ditl et al. 16 – squared vessel 150 mm × 150 mm: dp = f(t)  
 
Fig. 2. Experimental data from Ditl et al. 16 – squared vessel 150 mm × 150 mm: dp
* = f (N·t)  
 
90 
 
 
Fig. 3. Experimental data from Ditl et al. 16 – cylindrical vessel scale-up experiment: dp = f(t)  
 
 
Fig. 4. Experimental data from Ditl et al. 16 – cylindrical vessel scale-up experiment: dp
* = f (t)  
91 
 
 
 
Fig. 5. Experimental data from Ditl et al. 16 – cylindrical vessel scale-up experiment: dp
* = f (N·t)  
 
Fig. 6. Experimental data from Ditl et al. 16 – model parameters: dpf  = f (P/(·Vimp)); cylindrical 
vessel – four different vessel sizes, square vessel – four different impeller speeds 
92 
 
 
Fig. 7. Experimental data from Ditl et al. 16 – model parameters: k* = f (N·d0.89); cylindrical vessel – 
four different vessel sizes, square vessel – four different impeller speeds 
The following dimensional relationships were found for model parameters dpf and k
*
: 
 
41)(62331 /
imppf
V/(P.d  (R2 = 1),  (10) 
 
36728907 )(101781 ..* dN.k   (R2 = 0.721), (11) 
where  
 dpf  [m],  k
*
  [-], (P/(·Vimp)) [W/kg], N [RPS] and impeller diameter d [m]. 
 
As expected, the model parameter dpf is proportional to the mean dissipation rate in the 
impeller zone (P/(·Vimp))
-1/4
 as follows from Equation (9). The dimensionless rate-constant 
k
*
 was found to be proportional to shear strength on impeller surface ((N·d)0.89). The 
obtained relationships are graphically shown in Fig. 6 and 7.   
5.  Conclusions    
The following conclusions can be made: 
A simple dispersion model enabling the prediction of particle size changes over time 
was proposed. The model takes into account the type of breakup mechanisms, the non-
homogeneity of the local turbulent energy dissipation rate in an agitated vessel and the 
effect of number of times the liquid passes through the impeller and the impeller zone.  
The model was successfully tested on the data published by Ditl et al. 16. In their 
study, the dispersion of TiO2 particles in liquid was investigated. 
93 
 
The dimensional relations dpf = 31.623·(P/(·Vimp))
-1/4 
and k
*
 = 1.178·10-7 ·(N·d0.89)2.367) 
were found for model parameters, where dpf  [m], k
*
  [-], (P/(·Vimp)) [W/kg], N [RPS] and 
impeller diameter d [m]. 
 
 
A c k n o w l e d g e m e n t 
This research has been supported by the Grant Agency of the Czech Republic project 
No. 16-20175S ‘Local turbulent energy dissipation rate in dispersion systems.’ 
 
 
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nd 
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Dispersion Kinetics Modelling

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