Title (Calibri font, size 14)

CFD-PBM simulation of turbulent drop break-up in a high-pressure static mixer Ioannis Bagkeris1, Vipin Michael1, Robert Prosser1, Adam Kowalski2 1 2

School of Mechanical, Aerospace & Civil Engineering, University of Manchester, Manchester, M13 9PL, UK Unilever R&D, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, CH63 3JW, UK Abstract: This work investigates the simulation of dilute, low-viscosity emulsions in a pilot-scale ACIP2 Sonolator using Computational Fluid Dynamics (CFD) and Population Balance Methods (PBM). Comparisons between the simulations and the experimental work of Ryan [1] show excellent agreement of the pressure drop (𝛥𝑃) and reasonable agreement of the Sauter mean diameter (𝑑32 ). The effect of increasing oil viscosity (𝜇𝑑 ) in the 𝑑32 − 𝛥𝑃 correlation is also correctly captured. The numerical and the theoretical 𝑑32 − 𝛥𝑃 correlations are found to differ strongly with the experimental findings. The assumption of droplet break-up in the turbulent inertial (TI) regime (inherent to both theory and to current-state breakage frequency models) is found not to apply in the regions of intense break-up. Keywords: PBM, QMOM, drop breakage, emulsion, turbulence, orifice, high-pressure homogeniser

Introduction Emulsions are commonly encountered in food and personal care products; their formation involves the mixing of ingredients to form a coarse pre-mix and then successively reducing the drop size, typically via mechanical energy input. The resulting drop size distribution (DSD) is a critical property of the emulsion, affecting rheology, stability and appearance of the final product [2]. Industrial practice is still largely based on empiricism, and achieving the desired DSD remains a challenge since the interplay between emulsion formulation and processing conditions is complex and often poorly understood [3]. Droplet break-up in turbulent flows is the result of interactions between the droplet and turbulent velocity fluctuations whose length-scale is comparable to the droplet diameter [4, 5]. The turbulent velocity fluctuations can be characterised by the second-order longitudinal structure function 〈[𝛥𝑣(𝑑)]2 〉, which indicates the energy per unit mass contained in eddies of size ~ 𝑑 [6]. For locally isotropic turbulence and droplets with diameters in the inertial subrange, Kolmogorov’s second similarity hypothesis yields: 〈[𝛥𝑣(𝑑)]2 〉 = 𝛽(𝜀𝑑)2⁄3

(1)

where 𝜀 is the dissipation rate of turbulent kinetic energy and 𝛽 is a universal constant [7]. Drops with diameters in the inertial subrange are said to break-up in the turbulent inertial (TI) regime [8]. The Kolmogorov – Hinze theory [4, 5] predicts the maximum stable drop size in a turbulent flow as 𝑑𝐾𝐻 ∝ (𝜎⁄𝜌𝑐 )3⁄5 𝜀 −2⁄5 , obtained by balancing the normal stresses 𝛥𝑝(𝑑) ≈ 𝜌𝑐 〈[𝛥𝑣(𝑑)]2 〉 against the Laplace pressure 4𝜎⁄𝑑, where 𝜌𝑐 is the continuous phase density and 𝜎 is the interfacial surface tension. This prediction is strictly valid for systems with 𝜇𝑑 ∼ 𝜇𝑐 . For systems where 𝜇𝑑 ≫ 𝜇𝑐 , the higher viscosity of the dispersed phase increases the time taken to deform the drop. Equating the droplet deformation time to the eddy turnover time 𝑡𝑒𝑑𝑑𝑦 = 𝑑2⁄3 𝜀 −1⁄3 gives 𝑑𝑊 ∝ (𝜇𝑑 ⁄𝜌𝑐 )3⁄4 𝜀 −1⁄4 , interpreted as the maximum stable drop diameter according to Walstra [8]. Using dimensional analysis, the turbulence energy dissipation rate 𝜀 may be replaced by 1⁄𝑧 (𝛥𝑃⁄𝜌𝑐 )3⁄2 where z is a typical distance over which the turbulence energy is dissipated and 𝛥𝑃 is the typical pressure drop in the mixer [1]. Thus in high-pressure mixers, the maximum stable drop size may be expected to scale with the pressure drop as 𝑑𝐾𝐻 ∝ 𝛥𝑃−0.6

(2)

for low dispersed phase viscosity systems and as 𝑑𝑊 ∝ 𝛥𝑃−0.375

(3)

for high dispersed phase viscosity systems. The Sonolator is a high-pressure homogeniser manufactured by Sonic Corporation. The main components of the device (Figure 1) consist of an inlet chamber and a main chamber housing a nozzle with an elliptic-shaped orifice, a knife-edged blade and an outlet chamber. The device operates by 16th European Conference on Mixing – Mixing 16

9-12 September 2018, Toulouse, France

forcing high pressure fluid through the orifice, and onto the fixed blade. High local rates of turbulence energy dissipation are encountered during this process, which promotes droplet break-up and turbulent mixing.

Methodology Two different oil viscosities and five different mass flow rates matching those from the experiments of Ryan [1] have been chosen for the simulation. The dispersed phase volume fraction (𝜑) ranges from 0.5% to 10%. Therefore 𝜑 is low enough for Non-Newtonian effects to be neglected [3]. Single phase simulations are carried out since the Stokes number (based on volumetric flow rate and droplet diameter) 𝑆𝑡 ≪ 1. In general, breakup mechanisms in turbulent dispersions can be classified into four main categories [9]: (a) turbulent inertial stresses, (b) turbulent viscous stresses, (c) shearing-off and (d) interfacial instabilities. Mechanism (b) does not apply here because the droplet size remains larger than the Kolmogorov length scale. Mechanisms (c) and (d) also do not apply because they are relevant only to large droplets whereas the droplet size in the studied system is range 3-33μm. Hence breakup due to turbulent inertial stresses is only considered. An additional mechanism found in the Sonolator (but not accounted for in the present study) is cavitation-induced stresses. Experimental results obtained with the Sonolator [1] suggest that cavitation does not significantly affect droplet size. To simplify the analysis, the effect of cavitation on the flow field and the droplet dispersion is neglected.

Figure 1: Model ACIP2 Sonolator main components: (a) elliptic-shaped orifice, (b) blade

Figure 2: Surface mesh of main components

The computational domain has been meshed with a structured grid; mesh independence is established via a number of metrics. The surface mesh of the main components is shown in Figure 2. The flow field is simulated using an Unsteady Reynolds-averaged Navier-Stokes (URANS) approach via the open-source software Code_Saturne [10]. Turbulence closure is achieved through the k-ω SST model [11]. No wall function is employed since the mesh is refined close to the wall. A second order upwind scheme is used to evaluate the convective fluxes and an implicit first order Euler scheme is used for time discretisation. The general population balance equation (PBE) for the number density function 𝑛(𝐿, 𝒙; 𝑡) of droplet with diameter 𝐿 is: 𝜕 𝜕 𝜕 𝜕 𝑛(𝐿, 𝒙; 𝑡) + 𝑈𝑖 𝑛(𝐿, 𝒙; 𝑡) − (𝛤 𝑛(𝐿, 𝒙; 𝑡)) 𝜕𝑡 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑥𝑖 ∞

= ∫ 𝑝𝑎(𝐿′ )𝑏(𝐿|𝐿′ )𝑛(𝐿′ , 𝒙; 𝑡)𝑑𝐿′ − 𝑎(𝐿)𝑛(𝐿, 𝒙; 𝑡).

(4)

𝐿

The integral on the right hand side of Eq. (4) is the birth term for new droplets of size 𝐿 formed due to breakage from droplets of size 𝐿′. 𝑎(𝐿)𝑛(𝐿, 𝒙; 𝑡) term is the death rate due to breakage; 𝑎(𝐿) represents the breakage frequency. 𝑏(𝐿|𝐿′ ) is the PDF of droplets breaking from diameter 𝐿′ to droplets of diameter 𝐿, and 𝑝 is the number of the daughter droplets. The PBE is solved using the quadrature method of moments (QMOM) [12, 13] – essentially an assumed PDF approach where only the moments of the distribution are transported. The quadrature approximation of the QMOM is:

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∞

𝑁

𝑚𝑘 (𝒙; 𝑡) = ∫ 𝑛(𝐿, 𝒙; 𝑡)𝐿 𝑑𝐿 ≈ ∑ 𝑤𝑖 𝐿𝑖 𝑘 𝑘

(5)

𝑖=1

0

where 𝑤𝑖 and 𝐿𝑖 are the weights and the abscissas of the quadrature, respectively, computed with the product-difference algorithm [14]. Applying the moment transformation (Eq. (5)) to Eq. (4) gives: 𝜕 𝜕 𝜕 𝜕 𝑚𝑘 (𝒙; 𝑡) + 𝑈𝑖 𝑚𝑘 (𝒙; 𝑡) − (𝛤 𝑚 (𝒙; 𝑡)) 𝜕𝑡 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑥𝑖 𝑘 𝑁

∞

𝑁

= ∑ 𝑎(𝐿𝑖 )𝑤𝑖 ∫ 𝑝𝐿 𝑏(𝐿|𝐿𝑖 )𝑑𝐿 − ∑ 𝐿𝑖 𝑘 𝑎(𝐿𝑖 )𝑤𝑖 . 𝑘

𝑖=1

0

(6)

𝑖=1

For QMOM of order N, 2N transport equations are required: N = 2 is used in this study. The breakage frequency is calculated using the model of the Alopaeus et al. [15] which was developed for liquid-liquid systems when droplet break-up falls into the TI regime. The breakage model also captures the stabilizing effect of increasing 𝜇𝑑 , and is written as: 𝑐2 𝜎 𝑐3 𝜇𝑑 𝑎(𝐿) = 𝑐1 𝜀 1⁄3 𝑒𝑟𝑓𝑐 (√ 2⁄3 5⁄3 + ). 𝜌𝑐 𝜀 𝐿 √𝜌𝑐 𝜌𝑑 𝜀 1⁄3 𝐿4⁄3

(7)

The constants used here 𝑐1 = 90, 𝑐2 = 0.16 and 𝑐3 = 0.2, which have been reported by Solsvik et al. [16] for oil-water systems. Binary breakage is assumed, and the daughter size distribution is modelled using the 𝛽-distribution model of Hsia and Tavlarides [17].

Results and Discussion Contours of breakage frequency and 𝑑32 at the near field of the jet are presented in Figure 3 for 𝑚̇ = 0.1𝑘𝑔𝑠 −1 and 𝜇𝑑 = 10𝑐𝑆𝑡. As the shear layers develop from the nozzle edges, turbulence production and dissipation increase; the turbulent velocity fluctuations seen by the droplets increase and, therefore, breakage frequency increases. Given the inlet and the outlet 𝑑32 values (33.4𝜇𝑚 and 2.99𝜇𝑚, respectively), it can be seen from the figure that 80% of breakage has occurred by ~ 3𝑑𝑒 (𝑑𝑒 = 2√𝐴⁄𝜋 is the orifice equivalent diameter) before the blade is encountered. Centreline profiles of 𝑑32 /𝑑32,𝑂𝑢𝑡𝑙𝑒𝑡 (Figure 4) show that the majority of breakage occurs before the droplets encounters the ̇ ratio. blade; ~ 97% of breakage for the lowest (𝜇𝑑 /𝑚)̇ ratio and ~ 50% for the highest (𝜇𝑑 /𝑚) Comparisons of 𝛥𝑃 between the experiment and the CFD results show that predictions agree to within 4% or less; 𝑑32 predictions are relatively weaker. Results of 𝛥𝑃 against 𝑑32 are plotted in Figure 5. The increase in slope with decreasing 𝜇𝑑 is correctly captured by the numerical predictions, although the actual predicted gradients are larger than the experimental ones. The oil viscosities examined here fall into the low viscosity regime, and therefore, from Eq. (2) we expect the exponent to be closer to -0.6. With increasing 𝜇𝑑 the exponent is expected to increase towards -0.375 (Eq. (3)). While the trend predicted from theory is reflected in both the experimental and the numerical results, the magnitudes of the exponent in the experiment are much smaller. Figure 6 shows axial profiles of the turbulence length scales and 𝑑32 for one flow rate and two oil viscosities. The inertial subrange is indicated via the (𝑙 ⁄6, 60𝜂) boundaries suggested by Pope [7], where 𝑙 is the integral length scale and 𝜂 is the Kolmogorov length scale. The range between the large and the small scales becomes large enough for the inertial subrange to manifest at ~ 1𝑑𝑒 downstream. The evolution of 𝑑32 shows that drop break-up does not occur in the TI regime at any location along the centreline. Downstream ~ 1𝑑𝑒 drop size falls into the dissipation range. This means that the underlying assumption of Eq. (1) in the derivation of the theoretical correlations (Eq. (2)–(3)) and the breakage frequency model (Eq. (7)) does not hold within the range of parameters examined in this study. The effect of the departure from the TI assumption to the 𝑑32 − 𝛥𝑃 correlation will be the objective of future research.

Conclusions It has been shown that combined CFD – PBM can predict with reasonable accuracy 𝛥𝑃 and 𝑑32 at the outlet of a pilot-scale static mixer. The simulation provides insight on the flow development and its influence on the evolving droplet size, highlighting the regions where the bulk of breakage occurs. Comparisons between the experimental and the numerical/theoretical 𝑑32 − 𝛥𝑃 correlations show that, while the decrease of the exponent with increasing oil viscosity is correctly captured, there is significant 16th European Conference on Mixing – Mixing 16


disagreement in the magnitude of the exponent. Theoretical correlations and breakage frequency modelling are directly linked with the assumption of breakage in the TI regime, which was found not to apply in the present study; future work will entail droplet breakage modelling across the entire energy spectrum as well as the effect of cavitation.

Figure 3: Breakage frequency (𝑠 −1 ) (left) and 𝑑32 (𝑚) (right) in the ZY (top) and ZX (bottom) plane at the near field for 𝜇𝑑 = 10𝑐𝑆𝑡 and 𝑚̇ = 0.1𝑘𝑔𝑠 −1

Figure 5: 𝑑32,𝑂𝑢𝑡𝑙𝑒𝑡 − 𝛥𝑃 power-law correlations from experiment [1] and the present study

Figure 4: Graph of normalised 𝑑32 along the axial line −1𝑚𝑚 ≤ 𝑍 ≤ 4𝑚𝑚 for different mass flow rates and oil viscosities

Figure 6: Graph of turbulence length scales and 𝑑32 along the axial line 0 ≤ 𝑍 ≤ 4𝑚𝑚 for 𝑚̇ = 0.1𝑘𝑔𝑠 −1 and two different oil viscosities

Acknowledgements This work was carried out with the financial support provided by the EPSRC and Unilever under EPS iCase MA-2015-00510. The authors acknowledge use of Hartree Centre resources. References [1] Ryan D. J., (2015), Ph.D. thesis, University of Birmingham. [2] McClements D., (2015), Food emulsions: principles, practices and techniques. [3] Walstra P., (1993), Chem. Eng. Sci., 48: 333–349. [4] Kolmogorov A. N., (1989), Publications of A. N. Kolmogorov, Ann. Probab., 17: 945-964. [5] Hinze J. O., (1955), AIChE J., 1: 289–295. [6] Davidson P. A., (2004), Turbulence: an introduction for scientists and engineers. [7] Pope S. B., (2000), Turbulent flows. [8] Walstra P., Smulders P. E., (1998), Modern aspects of emulsion science. [9] Liao Y., Lucas D., (2009), Chem. Eng. Sci., 64: 3389–3406. [10] Archambeau F., Mchitoua N., Sakiz M., (2004), IJFV, 1: 1-62. [11] Menter F., (1994), AIAA J., 32: 1598–1605. [12] McGraw R., (1997), Aerosol Sci. Technol., 27: 255-265. [13] Marchisio D. L., Vigil R., Fox R. O., (2003), J. Colloid Interface Sci., 258: 322-334. [14] Lanczos C., (1988), Applied Analysis. [15] Alopaeus V., Koskinen J., Keskinen K. I., Majander J., (2002), Chem. Eng. Sci., 57: 1815-1825 [16] Solsvik J., Tangen S., Jacobsen H. A., (2013), Rev. Chem. Eng., 29: 241-356. [17] Hsia A., Tavlarides L., (1983), Chem. Eng. Sci., 26: 189-199. 16th European Conference on Mixing – Mixing 16
