Effects of tip streaming on the prediction of droplet size distribution in the presence of dispersants during subsea blowouts

Chemical dispersants are one of the most commonly considered non-mechanical techniques in spill response (NRC, 2005). The application of dispersants is mostly adopted on the water surface. The first large-scale application of dispersants in deep water is Deepwater Horizon blowout. Approximately 3 million liters of dispersants was applied at ~1500 m depth in the Gulf of Mexico to lower the interfacial tension between oil and water and to promote formation of small droplets (Kujawinski et al., 2011; Ryerson et al., 2012). Although the chemical process by which dispersants work are generally well understood, the effects on the droplet formation from the blowouts (e.g. DWH blowout), especially on the formation of very small droplets, namely tip streaming, are limited.

Tip streaming was first observed by Taylor (1934) and in recent years the tip streaming phenomenon has been linked to the presence of surfactants (Eggleton et al., 2001). Tip streaming occurs when the droplet deforms into a conical shape and a stream of very tiny droplets is ripped off the tips of the droplet pole. Previous studies on tip streaming are mostly focus on single droplet deformation. De Bruijn (1993) studied tip streaming phenomenon in simple shear flows experimentally and found that tip streaming can occur if interfacial tension gradients develop. Anna and Mayer (2006) observed microscale tip streaming in a microfluidic flow and found that the tip streaming occurs due to accumulation of surfactants in the cone tip. Based on the experimental observations, Gopalan and Katz (2010) suggested that the modal of the smallest sizes (e.g. 1 μm peak) in bimodal distribution observed during surface oil spill experiments is due to the tip streaming phenomenon.

Previous modeling studies also focus on the tip streaming phenomenon of a single droplet. The theoretical work regarding the interfacial force and concentration is mainly developed by H.A. Stone (1990). The additional presentation of the surfactant is usually modelled by adding an additional source term. Due to the shear velocity at the interface, the surfactant is mostly accumulated at the rear tip; thus, the rear tipoff the oil droplet occurs because of much higher surfactant concentration in the tip than the rest of the oil droplet. Meanwhile, the different concentration of surfactant on the oil droplet introduces Marangoni stress (Van Puyvelde et al., 2002). Eggleton et al. (2001) used a nonlinear model for surface tension to study the tip streaming from a drop in the presence of surfactants. James and Lowengrub (2004) investigated the single drop in the presence of extensional flows. The concentration of the surfactant is related to the local interfacial tension by equations of state. They further concluded that the increases of local surfactant concentration could result in a local deviation from the dilute limit, which further changes the Marangoni force to cause motion. To our knowledge, no study has reported the overall effects of tip streaming on the droplet formation in subsea blowouts.

Therefore, the objective of the current study is to develop a conceptual module in the thoroughly validated formation model, VDROP-J, to account for the tip streaming phenomenon in the droplet formation process in subsea blowout. Calibration of the new scheme and model validations are performed in association with CFD modeling and the underwater oil jet experiments.

The VDROP-J model (Zhao et al., 2014a) combines the transient droplet size distribution model VDROP (Zhao et al., 2014b) with correlations for jet hydrodynamics. The model, VDROP-J, relies on conceptually moving a given volume of fluids downstream of a blowout orifice, and allowing the volume to be subjected to variable mixing energy (actually decreasing) and dilution (due to water entrainment from the surrounding water body). The model VDROP relies on solving the following population balance equation:

where n is number concentration (unit: #/L³) of droplets of diameter d_i (unit: L) at a given time t (unit: t). β(d_i, d_j) is the breakage probability density function (dimensionless) for the creation of droplet of diameter d_i due to breakage of droplets of (a larger) diameter d_j. g(d_j), (unit: /t) is the breakage frequency of droplets of diameter d_j (discussed later). For droplet breakup, the first term represents the birth of droplets d_i resulting from the breakup of droplets d_j, while the second term represents the death of droplets d_i due to breakup into smaller droplets. For droplet coalescence, Γ(d_k, d_j) is the coalescence rate (unit: L³/t) (discussed later). The first term on the second row of Eq. 1 represents the birth of droplets d_i as a results of coalescence events occurring between droplets d_k and d_j to form droplets of size of d_i, while the second term represents deaths of droplets d_i due to the coalescence of drops d_i with all other droplets (including droplets of size d_i themselves) to form larger droplets.

The breakage rate g(d_i) is given by:

where S_ed represents the cross section area of eddy-droplet (unit: L²), u_e is the turbulent velocity of an eddy (unit: L/t), u_d is droplet velocity (unit: L/t), n_e is number concentration of eddies (unit #/L³), E_c is the average excess of surface energy needed to form a pair of daughter droplets or a small and large droplets, this term also known as formation energy (unit: J), E_v is the resistance energy due to viscous forces within the droplet (unit: J), e is the energy of the turbulent eddy that would cause breakup of the droplet (unit: J), and c₁ is an empirical constant equal to 1.3 (Tsouris and Tavlarides, 1994). In the context of jets/plumes, the parameter K_b was obtained (Zhao et al., 2014a) by correlation with the momentum via the relation:

where ρ is the density of the discharged fluid including oil and gas (unit: M/L³), U is the exit velocity (unit: L/t), Do is the orifice diameter (unit: L), U represents the effective exit velocity (Johansen et al. 2013) and discussed later. Attempts to correlate K_b to dimensionless quantities, such as the Weber number, the Reynolds number, or a combination of the two were made, but did not provide good correlations.

The coalescence rate of bubbles/droplets is expressed as:

where S_ij represents the cross section area (unit: L²) of the collided droplets with sizes d_i and d_j, u_i and u_j are the velocities of droplets d_i and d_j, respectively, t_ij is the coalescence time (s), and τ_ij is the contact time (s) between the two droplets d_i and d_j. The adjustable parameter K_c took the value of 3.0 × 10⁻⁵ (Zhao et al., 2014a; Zhao et al., 2014b).

For fluid discharges that include liquids and gases (e.g., air, methane), the effective exit velocity and flow rate should be used (Johansen et al., 2013). We developed an expression based on the work of Johansen et al. (2013), and it is:

where U_E is the effective exit velocity (unit: L/t), ⁠, Q_L and Q_G are the liquid and air/gas flow rates (unit: L³/t), respectively, ρ_L and ρ_G are the liquid and gas densities (unit: M/L³), respectively, and A is the cross section area (unit: L²).

The tip streaming phenomenon occurs when a section of the parent droplet deformed into a conical shape, then a stream of tiny droplets is ruptured off the tips after the cone extends to thin threads (De Bruijn, 1993; Gopalan and Katz, 2010). After the tiny daughter droplets are ejected, the parent droplet could attain a stable shape until another ejection (Eggleton et al., 2001). To simulate an average droplet size change due to tip streaming phenomenon in the blowout, the formulation of tip streaming module is based on the mass change of the parent droplet due to the rupture of droplets from the tip:

where M_i is the total mass of droplets with diameter di and mass mi (unit: M), M_i = n(d_i,t)m_i; k_tip is the mass change coefficient (dimensionless); and J₀ represents the mass losing rate during the tip streaming phenomenon per unit time (unit: M/t).

With a similar concept of losing mass from the parent droplet, the mass change coefficient k_tip was formulated from related theories of mass transfer coefficient. It is believed that the tip streaming is caused by the interfacial flow on the droplet surface, which convects the surfactants towards the pole due to the shear stress (Eggleton et al., 2001; Stone and Leal, 1990). This leads to a high concentration of surfactants at the pole and sharp gradient of interfacial tension on the droplet surface. When the interfacial tension at the pole reaches certain critical value, resistance to breakup from the parent droplet collapse. The interfacial spreading velocity, U_IF, of the surfactants at the interface between two fluid phases was estimated by Davies and Rideal (1963):

where Δσ is the interfacial tension difference causing spreading (unit: N/L); μ is the dynamic viscosity (unit: Pa.s), subtitle c and d represent the continuous and dispersed phase. Using Eq. 8, Handlos and Baron (1957) correlated the mass transfer coefficient due to the maximum effect of interfacial turbulence:

where Δσ is the maximum possible different of interfacial tension due to the presence of surfactants (unit: N/L).

Considering also the flow hydrodynamics and the diffusion of surfactants on the drop surface, the formulation of k_tip is expressed as:

where Re_d is the droplet Reynolds number, Re_d = u_dd_i/v_c ; D is molecular diffusivity of the surfactants (unit: L²/t). The tip streaming is definitely related to the interfacial tension reduction. Studies also show that the ratio of internal to external viscosity affects the occurrence of tip streaming phenomenon (Janssen et al., 1994; Tretheway and Leal, 1999).

The mass losing rate, J₀, could be the maximum possible mass rate that the daughter droplets are ripped off from the parent droplet, which is yet to be determined experimentally. Here, a modeling study by Computational Fluid Dynamics (CFD) was conducted to provide a rough estimate of J₀. The simulation was performed on the assumption that the material properties maintain constant in one cell in the domain and we used FLUENT via Volume of Fluid (VOF) method to track the interface under the effect of surfactants.

The convection-diffusion equations on the interface can be described as (Wang et al., 2014):

where Γ is the surfactant concentration at the interface (unit: M/L²); is the tangential velocity along the interface (unit: L/t); D_s is the diffusion coefficient along the interface (which is a constant in the present simulation) (unit: L²/t); ΓS_Γ represents the source term due to the net adsorption of surfactant at the interface, expressed as:

k_a and k_d indicate the rate constant of adsorption and desorption (unit: 1/t), respectively. The term C_s represents is the concentration of surfactant in the bulk that immediately adjacent to the interface and Γ_∞ is the maximum (saturated) surfactant concentration at the interface. k_aand k_d can be assumed as constant values when the absorption and desorption energy (E _a,d) are small compared to the product of k_b and T, where k_b is the universal Boltzmann constant (He et al., 2015).

We assumed the following relationship between interfacial tension and surfactant concentration:

Figure 1 shows a predicted series of occurrence of tip streaming for a droplet with 1 mm diameter. We assumed that the droplet deformation is axisymmetric; thus only half of the droplet evolution is shown in Figure 1. It is also assumed that the oil and surfactants are premixed and the droplet is rising by its own buoyancy. With continuous rising, the droplet is deformed into a conical shape as observed by previous experimental studies. The tip of the cone continues stretching with time into a thin thread until a tiny droplet is ripped off from the tip of the thread. The predicted process of tip streaming is consistent with the experimental observations. Based on the simulation results, we estimated that the volume losing rate is in the order of ~10⁻¹² - ~10⁻¹⁷ m³/s. Using an oil density of 800 kg/m³, the J₀ could be in the range of ~10⁻⁹ - ~10⁻¹⁴ kg/s.

The tip streaming module was incorporated in the VDROP-J model. The mass was conserved at each time step during the simulation; that is, the volume of parent and daughter droplets after the breakup due to tip streaming was re-distributed into the corresponding size bins. It is observed that the shed daughter droplets usually had radii 2 orders of magnitude smaller than their parent droplets (De Bruijn, 1993), which gives ~1 μm daughter droplets from ~100 μm parent droplet. Gopalan and Katz (2010) also reported that droplets as small as 2.8 μm was produced from the breakup of a thread section in the quiescent case. Therefore, in the current study, all the daughter droplets from tip streaming were assigned into size bins less than 3 μm.

Experiments of underwater oil jets performed by Belore (2014) were reproduced from VDROP-J model. The Dorado crude oil was used in the experiments with a density of 875 kg/m³. Due to the concern of oil viscosity measured by Belore (2014), the viscosity of 4cp reported by Hearn (1972) was used in the simulation. Experiments were conducted in Ohmsett wave tank in Leonardo New Jersey. The tank is 20 m wide, 200 m long, and filled with sea water up to about 2.4 m depth. Oil, air, and dispersants were released through a 4.5 mm horizontal orifice. For the case simulated in the current study, the flow rates for oil and gas were 1500 and 8280 mL/min, respectively, which gives a GOR=5:1. The LISST (laser in-situ scattering transmissometry) was used during the experiments to obtain the droplet size distribution in the discharged plume. The LISST was placed 1 m above the exit.

Figure 2 shows the results for the case of DOR (dispersant to oil ratio)=1:50. The interfacial reduction (IFT) of 1000 fold was assumed in this case based on data reported by Brandvik et al. (2013). Without considering the tip streaming phenomenon (Figure 2a), the basic profile of the size distribution was generated by VDROP-J model, comparing with the experimental data, with a relatively large volume in some of the size bins (e.g. 30–50 μm, 200 μm). Note that elevated volume in the smallest two size bins (3 and 4 μm) was observed from the measurements; however, it was not predicted by the VDROP-J model without tip streaming module. For the case with the consideration of tip streaming (Figure 2b), the simulation results were largely improved by the tip streaming module and match well with the experimental data for most of the sizes. Because all the daughter droplets from tip streaming were distributed into size bins less than 3μm, the predicted volume in 3μm bin size is larger than the measurements in that size bin. But in reality, the tip streaming droplets could be in different sizes larger than 3 μm. This could be improved in accordance with experimental and numerical investigations in the near future. The device LISST may also capture only portion of the volume of the out of range particles (less than the lower detectable size 2.5 μm), which will be discussed later. The calibrated J₀ is 5.0 × 10⁻⁷ mg/s, which is within the range estimated from our simulations (~10⁻⁹ -~10⁻¹⁴ kg/s, see Section 2).

Results of increase of DOR to 1:200 are shown in Figure 3. The IFT reduction was assumed to be 5 fold (Abdelrahim, 2012). Modeling results match well with the experimental data. Also, due to the less amount of dispersants in the system (small Δσ in Eq. 10), tip streaming phenomenon is not occur, which is consistent with the observations, where there are no elevated volume in the smallest size bins. The differences between modeling and experimental data for the two DOR cases (Figure 2 and Figure 3) could also rely on the air bubbles inside the discharged plume. Though most of the air bubbles could leave the plume quickly after the release, there may be some small ones remain in the plume and captured by the LISST, causing a relatively large volume in some of the large size bins.

Due to the leakage of signal outside the filter pass-band, the size distribution in the smallest size bins from LISST measurements could be affected by the droplets smaller than the detectable limit (e.g. 2.5 μm for LISST-100X). Figure 4 shows the LISST measurements of spherical particles with a diameter of 0.5 μm. Leakage occurs in the smallest two size bins, with 95% of the volume located in the bin of 2.73 μm. However, the LISST only captured about 20% of the input volume (0.66 μL/L). For the cases predicted in the current study (Figure 2 and 3), the low DOR (Figure 3) clearly shows that the water in the tank did not contain small particles (no volume detected in the small bins). The modal of the smallest bins in the high DOR case (Figure 2) demonstrates that there were small droplets in the bins of 3 and 4 μm, or even smaller, and more likely the latter case. Thus, the formation of these small droplets was most likely due to the tip streaming phenomenon as we presented in this paper.

The effects of tip streaming on the prediction of droplet size distribution in the presence of dispersants in underwater oil jets were investigated in the current study. The thoroughly calibrated VDROP-J model was used for the simulation. A new conceptual module to account for the tip streaming phenomenon when dispersants are present was developed based on the mass change of the parent droplet. The VDROP-J model was then validated with experimental data of underwater oil jets. Results show that the new module for tip streaming largely improved the predictions for the case with relatively high DOR where tip streaming could occur. Further investigations are still needed both experimentally and numerically to calibrate the parameters and improve the modeling efforts for cases with the addition of dispersants.

When subsurface dispersant injection is used during a subsea blowout, the new model development would provide valuable information on droplet formation for decision makers and research groups. Addition of dispersant could promote large amount of micron or sub-micron droplets due to tip streaming process. Quantifying these small droplets is essential for estimating the oil remained in the deep ocean and dissolved in the water column, and also the natural biodegradation process of low solubility oil components.