Dynamic simulation of emulsion formation in a high pressure homogenizer

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Abstract

A simulation model for emulsification in high pressure homogenization (HPH), based on a population balance approach, is developed assuming it to be controlled by three simultaneous processes; fragmentation, coalescence and adsorption of a macromolecular emulsifier. The aim is to investigate the implications of adding a set of models together; studying the effects of dynamics, size effects and process interactions.

For fragmentation, turbulent inertial and turbulent viscous forces are included using a dynamic model based on the Weber and Capillary number. It was extended to include a deformation time scale.

The rate of adsorption and coalescence is assumed to be controlled by the collision rate of macromolecular stabilizer and bare interface, modeled using convective and diffusive transport in turbulent flow.

By comparing simulation results to general trends found in the literature, it can be concluded that the models can reproduce the general HPH process well. By dividing the active region of emulsification in the homogenizing valve into discrete steps, the dynamic process could also be examined, indicating the homogenization process being composed of three stages with coalescence predominantly found in the last one.

Introduction

High pressure homogenization (HPH) is used in chemical, pharmaceutical and food applications in order to form emulsions with small drops and narrow size distributions. Size and distribution width of the drops are important for many factors including stability, reactivity and mouth-feel.

In a high pressure homogenizer the emulsion is forced through a narrow gap (order 10–100μm) by applying a large pressure gradient (10–100MPa). Fig. 1 shows a highly schematic and not to scale representation of the gap region. Pressure is controlled by adjusting the position of the forcer, expanding or contracting the gap height, and measuring the pressure loss over the gap. Drops will break-up and form a finer emulsion when passing through the active region of emulsification found at the gap outlet (Innings and Trägårdh, 2005) as indicated in Fig. 1.

In order to link homogenization result, often described by the Sauter mean diameter, d32, to operating parameters some authors have worked on obtaining empirical correlations based on a large number of experiments (Phipps, 1975; Floury et al., 2004a). This approach has resulted in different expressions, each describing the dependency of the parameters differently without much possibility of quantitatively describing the reason for this. However, some general trends are similar between experiments. One of the most well-studied dependencies is that for homogenization pressure:d32ΔPq1where q1<0 is an empirical constant. It is generally believed that the value of the constant is closely linked to the fragmentation mechanism (Walstra and Smulders, 1998). The exact nature of this fragmentation is somewhat disputed. Among the most common suggestions are elongation in the inlet due to high velocity gradients, laminar shear (Walstra, 1993), turbulent shear forces in the gap and/or the outlet (Mohr, 1987; Walstra and Smulders, 1998) and cavitation (Kurzhals, 1977, Kiefer, 1977). There is no consensus on the various mechanisms and how they vary between different geometries and emulsifier systems but if small laboratory homogenizers are excluded there is a quite general agreement on turbulent forces as the major contributor (Phipps, 1985; Walstra and Smulders, 1998; Floury et al., 2000). The same conclusion was drawn by Innings and Trägårdh (2005) by drop visualization in a scale model of the homogenization gap region.

A second parameter of importance for fragmentation is the dispersed phase viscosity, μD. An exponential dependency of the same type as Eq. (1) is often found experimentally:d32μDq2The constant q2 in Eq. (2) is found to be positive (Walstra, 1975, Pandolfe, 1981). This is commonly explained by the dependency on drop deformation time. A drop with a higher viscosity will need a longer time for deformation when a given force acts upon it. If this time is of the same order or lower than the relevant time scales for the fragmentation mechanism, break-up will be partially controlled by this deformation process (Pandolfe, 1981).

Based on the experimental findings of homogenizing pressure and dispersed phase viscosity, turbulent fragmentation can be seen to be controlled by two factors (Walstra and Smulders, 1998); the amount of dissipated energy, closely linked to the pressure, and the drop deformation time relative to the turbulent eddy life time, linked to disperse phase viscosity. This motivates the modeling of fragmentation frequency as a combination of the two factors. A similar method was used by Tjaberinga et al. (1993) in modeling fragmentation of drops in a mixing pipe.

Other parameters known to affect the result to a significant extent is the volume fraction of oil and the amount and type of emulsifier. Experiments reveal (Taisne et al., 1996; Narsimhan and Goel, 2001, Floury et al., 2004b) that severe (re)coalescence can be found and that the extent is closely linked to homogenization pressure and emulsifier. As coalescence is a second order process, whereas fragmentation is usually considered first order with respect to drop concentration, a higher volume fraction of oil will increase coalescence relative to fragmentation.

Emulsifiers acts partly by lowering surface tension, and thus the stabilizing Laplace pressure, see Eq. (10). Alternatively they can alter the rigidity of the surface (Jones and Middelberg, 2003). Both processes increase fragmentation. They also hinder coalescence by adsorbing at the interface. Different explanations of the specific mechanism have been suggested. Most commonly found is repulsive forces due to steric and/or electrostatic interactions and dynamic adsorption processes such as Gibbs–Marangoni effects (Walstra and Smulders, 1998; Narsimhan and Goel, 2001).

In this study we focus on macromolecular emulsifiers which we define as emulsifiers showing a convective rather than diffusion controlled adsorption process with low surface pressure and surface viscosity. For macromolecular emulsifiers, adsorption at the newly formed interface display kinetics comparable to the time of the homogenization. Nilsson and Bergenståhl (2006a) have shown that the time scale of adsorption is comparable to that of coalescence, revealing the importance of including adsorption phenomena in modeling high pressure homogenization.

Fragmentation, coalescence and adsorption are to a large extent determined by the pressure–velocity flow field in the active region. Experimental flow field measurement of high pressure homogenization is challenged by the small geometrical scales and large velocity gradients. The pressure field over the gap was measured by Phipps (1974) by mounting pressure sensors on the gap, but the technique is highly intrusive raising questions about how representative they are for normal homogenizer equipment. Innings and Trägårdh (2005) used particle image velocimetry (PIV) on a carefully scaled model but was unable to obtain a very high resolution of measured velocities.

Others have used computational fluid dynamics (CFD) in order to calculate the flow field (Stevenson and Chen, 1997, Kleinig and Middelberg, 1996, Kleinig and Middelberg, 1997; Floury et al., 2004a, Steiner et al., 2006). Although promising in theory, CFD is limited for example by imperfections in the turbulence modeling and also in only model one phase flow, not taking disperse phase modulation into account. Even more critical is the geometrical difference between the homogenizer in Fig. 1 and the ones considered in the CFD studies. The forcer in Fig. 1 slopes outwards after the gap. This will lead to an increase in curvature of the stream lines and thus lower the validity of the Boussinesq hypothesis (Pope, 2000) used in the CFD simulations above.

The conclusion from the survey above as well as by Walstra (1993, p. 341) is that high pressure homogenization can be described by three simultaneous processes: fragmentation, coalescence and adsorption. Despite this most studies only include one of the mentioned processes. For example Mohr (1987), Pandolfe (1981), Phipps (1985), Walstra and Smulders (1998) and Vankova et al., 2007a, Vankova et al., 2007b, focus on fragmentation while Mohan and Narsimhan (1997) only model coalescence. Rarely two processes are combined such as by Nilsson and Bergenståhl (2006a) with coalescence and macromolecular adsorption. All of the abovementioned studies only model or discuss the resulting mean drop size, d32. In many cases the shape of the distribution is of great importance, both theoretically and technically as has been shown by Walstra and Oortwijn (1975) for the distribution width on emulsion creaming stability. The models described above also yield only an estimation of final size without giving any insight into the dynamic process.

In order to describe the full development of drop size distribution over time the population balance equation (PBE) can be used. The PBE is an integro partial differential equation of conservation law type (Ramkrishna, 2000). This approach has been used by Soon et al. (2001) for a jet homogenizer and Vankova et al., 2007a, Vankova et al., 2007b for a narrow-gap homogenizer but then by only considering fragmentation. A similar method was used by Reddy et al. (1981) to simulate flocculation. In this study all three processes are considered simultaneously.

In this paper we focus on first to obtain a closed set of models for describing the fragmentation, adsorption, coalescence and turbulence in a high pressure homogenizer. Secondly, based on this set of rather simplistic models for the separate processes; integrate these in a dynamic PBE framework to see how well we are able to reproduce some of the characteristics of the emulsification process; thus examining the effects of dynamics and interaction between different processes. This should be seen as a first step in developing a full and functioning simulation model for the homogenization process including all relevant processes. In a recent paper (Håkansson et al., 2009) we discuss the impact of the operating parameters using the simulation model and also give a qualitative representation of the turbulent jet responsible for the flow field.

The model is intended for studying the dynamic emulsification process and to examine the relation between fragmentation and coalescence at different positions in the active region and see how well this corresponds to experimental correlations. Development of a model could be an important way of gaining insight into the linkage between geometry and the spatial emulsification process which in turn is of outmost important in high pressure homogenizer design.

Section snippets

Population balance equation

Consider a polydisperse emulsion as a drop volume distribution described using a distribution function n(v,t). This should be interpreted such that the emulsion contains n(v,t)dv drops with drop volume in the interval [v,v+dv] at time t in a unit volume of dispersion.

Consider a number of drops of volume v at time t and this quantity will increase and decrease with the fragmentation and coalescence in the following four ways.

  • (i)

    The number of concentration of drops, n(v,t)dv, increases when larger

Model integration and solving

The adsorption rate from Eq. (33) affects the system implicitly by altering the coalescence efficiency in the coalescence frequency expression, β. Flow models are used to set the dissipation rate that effects each term and an integration time for each step.

Discretization of Eq. (7) is generally needed for obtaining a solution. The Fixed Pivot Technique of Kumar and Ramkrishna (1996) was used for the coalescence term and a direct discretization as used in Spicer and Pratsinis (1996) for the

Evaluation

In order to solve the system of models discussed above, numerical values on constants and physical parameters must be decided on. The physical parameters in the simulation results can be found in Table 1. The values are chosen in order to describe an oil-in-water emulsion stabilized by macromolecules or proteins (dE) homogenized in a pilot scale homogenizer (Q,ri,re).

Resulting size distributions are presented as volume fraction distributions. Fig. 3 gives some examples of simulation results for

Notation

c,Cmodeling constants
CaCapillary number
cEconcentration of emulsifier in continuous phase, kg/m3
csdistribution width
ddiameter of drop, m
d32Sauter mean diameter, m
dEhydrodynamic diameter of macromolecular emulsifier, m
Emean energy content of flow, W
fdaughter drop size distribution function
gdrop break-up frequency, s-1
Gvelocity gradient due to turbulent viscous shear, s-1
hgap height, m
kBBoltzman's constant, J/K
K,Kmodeling constants
leEddy size, m
mnumber of fragments formed per break-up of a drop
M

Acknowledgments

This work was financed by the Swedish Science Council (VR) within the project “Chemical and Hydrodynamic Control of Break-Up and Coalescence during the Emulsification Event in a High-Pressure Homogenizer”.

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