High pressure homogenization is a method of emulsification that is widely used in chemical, pharmaceutical and food applications in order to form emulsions with small drops and narrow size distributions. Modeling and optimization of a small homogenising unit are often restrained by a lack of information of the flow condition within the homogenizer chamber. As detail visualization of flow in such a small chambers is not often possible, a numerical investigation of flow within homogenizer chamber is presented using a commercial Computational Fluid Dynamics (CFD) code. Results of simulation are obtained using the Finite Element Method (FEM) and a Standard k-ε turbulence model with near wall treatment conditions. The results are compared with the results of Floury et al. who have done simulation based on Finite Volume Method. The comparison showed good agreement between present work and theirs. More detailed information on the comparison are investigated.
Keywords: homogenization, numerical simulation, CFD, Pressure, turbulence
1.Introduction
Emulsions are dispersions of two or more immiscible liquid phases usually water and oil, one of which is dispersed as droplets in one another [1,2,3,4]. Size and distribution width of the drops of emulsions affect on many factors including stability, reactivity, rheology, appearance, color, mouth-feel, shelf life and texture [1,3,5,6]. The aim of homogenization is refinement of the droplets size as much as possible (often to obtain drops smaller than 1 μm) and narrowing the droplets size distribution (DSD). The most frequently used systems of emulsion production are rotor-stator systems, high-pressure systems, membrane systems and ultrasonic systems [7,8]. Detail information on the homogenising systems can be found in [6,7,9,10,11,12,13,14], but generally it is well known that rotor-stator systems can not produce emulsion with DSD < 1 μm because their energy density is lower but their dispersing zone is larger [6]. In one hand, beside ultrasonic and membrane systems are capable of producing sub-micron emulsions but they are efficient for laboratory scale not industrial production [1,7,15,16]. On the other hand, in high-pressure systems DSD as small as 0.1 micrometer can be produced with high throughputs [5,6,7]. The energy densities needed for the production of fine emulsions (drops of 0.2 μm) can generally only be achieved in high-pressure systems [7].
There is no general consensus on the mechanisms of drop breakage in the homogenizer, but studies show that the most common ones are shear stresses, extensional flow stresses, turbulent stresses, inertial forces and cavitation stresses [17,18,19,20].
High pressure homogenizers that consist of a high-pressure piston pump and a narrow gap, are the most efficient means for producing sub-micron emulsions in low viscous fluids, as the pump creates a pressure of 100-700 MPa, and the emulsion is accelerated to velocities of up to hundereds of m/s in the gap [17,21,22,23]. These systems that are classified based on the nozzle geometry, design and flow guidance can be subdivided into radial diffusers, jet dispersers, Microfluidizers, and orifice valves [6,7]. A type of radial diffuser system that is called Stansted high pressure homogenizing valve is simulated in this work. Referring to Fig. 1 that shows the schematic of this homogenizing valve, the high pressure fluid is fed axially into the valve seat, and then accelerated radially into the gap between the valve and seat, and at the end leave the valve seat at atmospheric pressure. When fluid leaves the gap, it becomes a radial jet that stagnates on an impact ring before leaving the homogenizer.
Stansted high pressure homogenizing valve was simulated by Floury et al. [24], in this work it is chosen too for simulating flow pattern. Floury et al. used Fluent CFD package based on Finite Volume Method while here simulation is done using Comsol CFD package based on FEM. Like original work by Floury et al., fluid flow is assumed to be single phase, Newtonian, isothermal, incompressible and steady-state. Some hydrodynamical parameters like velocity profile, pressure,turbulense intensity, and streams are obtained in four different pressure and some of them are compared with the results of Floury et al.
Fig. 1. Schematic view of the Stansted high pressure homogenizer
2. Simulation
2.1. Valve Geometry
Valve gap height (h) is dependent on the operating pressure and can not be measured directly, but based on Phipps derived equation [24], when the flow rate is known (Q=10 l/h), it is possible to approximate h in each run. As calculation of h by Floury et al. for four operating pressures of 26, 92, 225, and 340 MPa resulted in 4.74, 3, 2.3, and 2 μm respectively, so these values of h are used in this work. As the other dimensions of the homogenizing valve in the upstream and downstream part of valve are known, the geometries for four different condition are easily implemented. As valve geometry is axially symmetric, instead of 3 dimentional simulation, 2 dimentional simulation is applied.
2.2. Flow Regime
For determining the flow regime in the valve, Reynolds number must be calculated. As the hydraulic diameter of the annular gap is dh=2h, calculation of Reynolds number in the upstream part of valve, gap, and downstream part of valve (with Ï=1000 kg/m3 and μ=10-3 Pa.s) even at the rigorous condition are 315, 550, and about 2000 respectively (transition from laminar to turbulent occurs at a Reynolds number of about 2000). It can be seen that the flow regime is laminar or transational in the valve, but it is expected that coming out flow from the gap will expand, causing regions of eddy formation [24], so a turbulence model (Standard k-ε model) is added to simulation equations.
2.3. Governing Equations
Continuity and momentum equations are the base equation for CFD simulation and calculating hydrodynamical variables. As it is assumed to use a turbulence model, Reynolds averaged equations are the main equation, on the other hand, due to assumption of Newtonian, isothermal, incompressible, single-phase, steady state and constant density and viscosity the Reynolds averaged Navier-stokes equations will be:
(1)
(2)
While RNG k-ε turbulence model was chosen by Floury et al., in the present work, Standard k-ε turbulence model is chosen to couple with Reynolds averaged Navier-Stokes equations.
(3)
(4)
(5)
with Cμ = 0.09 , Cε1 = 1.44 , Cε2 = 1.92 , σk = 1.0 and σε =1.3
where k is the turbulent kinetic energy and ε is the dissipation rate of k.
2.4. CFD Simulation
The numerical procedure is based on a FEM with unstructured mesh [25,26]. Lagrange Elements used for approximation of the solution. Meshes that were unstructured, were gradually refined to reach a solution that does not change any more by increasing finite elements number. Due to many degrees of freedom it was supposed to use iterative solvers for resulted linear systems.
The convergence of the results like Floury et al. work, checked according to two criteria. First, all the normalized residuals must be less than 10-6 (instead of using 10-5 as Floury et al.). Secondly, iteration of solution do not change the simulation results [26].
2.5. Boundary Conditions
In the previous studies [27,28] a constant velocity value was imposed at the inlet of valave, while Floury et al. used constant relative pressure at the inlet. As it is well known, to have a well posed problem with Navier-Stokes equations, velocity type boundary condition should be applied at the inlet. So a combined approach is used at the inlet, first velocity type boundary condition was applied, when solution converged, boundary type is changed to pressure and turbulence intensity was set at 1%. At the outlet of the valve, atmospheric pressure is assumed.
Turbulence close to a solid wall is very different from isotropic free-stream turbulence, so this must be accounted for a proper model [29]. In this approach, which is used by these application modes, an empirical relation between the value of velocity and wall friction replaces the thin boundary layer near the wall. Such relations are known as wall functions.
2.6. Logarithmic Wall Functions
Logarithmic wall functions applied to finite elements assume that the computational domain begins a distance δw from the real wall. They also assume that the flow is parallel to the wall and that the velocity can be described by
(6)
Here U is the velocity parallel to the wall, uτ is the friction velocity defined by
(7)
k denotes the Kármán's constant (about 0.42), and C+ is a universal constant for smooth walls. Further, l* is known as the viscous length scale and is defined by
(8)
The logarithmic wall functions are formally valid for values of between 30 and 100. For high Reynolds number, the upper limit can be extended to a few hundred [29].
3. Results and Discussion
Accrding to previous studies [24,27], an extream pressure decrease is expected as the fluid inter the gap and the flow velocity increase. Fig. 2 shows the pressure along the line that connect inner wall of external cylender of valve-upstream part to the end of gap (this line can be seen as dashed line in Fig. 3). The pressure result of Floury et al. is shown in Fig. 3. Comparison of Fig. 2. And Fig. 3 shows that a sudden pressure decrease along the gap, that agrees with Kleining et al. results too [27].
Fig. 2. Absolute pressure in the valve gap area.
Fig. 3. Absolute pressure in the valve gap area [24].
As operating pressure increase,when fluid leave the upstream part of the valve and flows in the gap, the velocity increases rapidly (Fig. 4), because cross section of flow decrease suddenly. So this extentional flow generates intense velocity gradients that have major effect on droplets disruption. Fig. 4 shows the volicity near gap inlet at upstream part of valve, while Fig. 5 is the Floury et al. results at the same radius and same condition.
Fig. 4. Velocity at the valve-upstream near the gap as a function of operating prssures.
Fig. 5. Velocity at the upstream near the gap as a function of operating prssures [24].
On the other hand, at a fixed operating pressure, as the flow cross the gap and the radius of gap decrease, velocity increase, due to decreasion of cross section of fluid flow (Fig. 6).
Fig. 6. Velocity profile at the gap (with pressure of 340 MPa) as a fuction of gap height in different radious.
4. Conclusions
The present results show a good agreement with the results of Floury et al., expressing the homogenizer valve is simulated based on FEM method using k-ε Standard turbulence model approach. Although the briefness of this report did not allow to show the turbulence characteristics of flow, but they are obtained and show similarity to the original work results.
The variables that are discussed here, beside giving detailed information on fluid mechanic and definning functions of disruption of droplets, are necessary for the next stage that is modeling of particle size distribution of drops, because the variables like turbulence kinetic energy are essential as a part of breakage functions.
At the end, it should be noticed that some assumptions about flow like, being single phase, Newtonian and isothermal are imposed. May it is better to couple the energy balance equation to primary equations set for more reliable simulation.
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