Coalescence During Chemical Dispersion of Oil Spills

Amanda Virkus

Supervised by Dr.  Brian Wrenn
Department of Electrical and
Systems Engineering
University in St. Louis
Spring 2007



I have worked in the Environmental Biotechnology Laboratory under Dr. Brian Wrenn from Fall 2005 – Spring 2007.  During the course of my research, I studied various facets of chemical dispersion of crude oil.  Chemical dispersion is an oil spill response technology that can be used in cases when mechanical recovery is expected to be ineffective or too slow to protect sensitive resources.  Dispersing an oil spill effectively removes oil from the water surface and entrains it into the water column below.  Due to the fact that the oil is not removed from the environment, there are concerns that marine and benthic communities can be adversely affected by dispersion.  However, in some open-water environments, water column resources are less abundant and the oil can be diluted to nontoxic levels very quickly.  Therefore, surface-dwelling organisms, such as marine mammals and birds, can be protected with little harm to other compartments.  In addition, because the oil is reduced to small droplets, chemical dispersion might enhance its biodegradability.  Although dispersion has been used for over 30 years, it is still not completely understood.




In an attempt to further our understanding of oil dispersion, I studied the coalescence of entrained oil droplets.  The droplets that are formed during dispersion could coalesce during further mixing to form larger droplets which may resurface.  A discrete model of coalescence kinetics (1) was created by Smoluchowski [1]:




where Nk /∂t is the rate of change in the number of droplets of volume vk due to coalescence, α represents the collision/sticking efficiency (the fraction of effective collisions), β(vi , vj) represents the frequency of collisions between droplets of volume vi and vj (where vi + vj = vk), and ni is the number of droplets of volume vi.  The first term represents the formation of droplets of size vk from the coalescence of smaller droplets.  The second term represents the loss of droplets of size vk from the coalescence of droplets of size vk with other droplets. 


Most of the values in this model can be easily determined.  The number distribution of droplets of various diameters is obtained using an optical particle counter (OPC).  The frequency of collisions term, β(vi , vj), can be determined by considering Brownian motion, shear, and differential sedimentation.  With our data, Brownian motion can be ignored because it affects particles with diameters less than 1μm [4] and the OPC used in our experiments can only measure droplets with diameters greater than or equal to 2μm.  Therefore, the frequency of collisions is a combination of turbulent shear and differential settling.




where frequency of collisions due to turbulent shear can be determined using the following model [1]:




and the frequency of collisions due to differential sedimentation can be determined using [1]:


                                                            .                                      (4)


However, the sticking efficiency, α, must be estimated.


Project Progress


All bench-scale oil dispersion experiments performed in the Environmental Biotechnology Laboratory are analyzed by determining the particle-size distribution.  In this lab, an optical particle counter (OPC) is used for this task.  This particle counter can determine the number concentration of droplets, n (ml-1), of dispersed oil droplets in the size range 2 μm to 125 μm.  The OPC counts the number of droplets in fifteen user-defined size bins.  Therefore, each droplet-size distribution is described by fifteen discrete data points (Table 1). 


However, when the Smoluchowski model of coalescence is applied to data described only by fifteen different droplet sizes, the issue of conservation of volume arises.  Therefore, my first goal was to fit a continuous function to these data.  This would make volume conservation very easy.  Since the droplet-size distributions had several major modes, we decided to fit a function consisting of the sum of three lognormal distributions of the form:



to the multimodal discrete data set using the Levenberg-Marquardt algorithm [2,3].  An example set of volume distribution data is shown below in Figure 1.  The third mode in this data set is quite small.  Some experiments show more evident modes, and after continued mixing and droplet coalescence, these modes become more pronounced.  The graph in Figure 1 contains both experimental data and a sum of three lognormal distributions to illustrate how this type of function would be a good fit for the data. 

Figure 1: Sum of lognormal distributions fit to experimental data


However, this optimization procedure never converged to a solution.  After various attempts to fix this problem, I eventually abandoned the attempt.  It is possible that the LM algorithm does not work with this sort of model and/or data set, or that we may need to transform our data set in some way.


After failing to fit a continuous curve to our laboratory data, I implemented the discrete Smoluchowski coalescence model in MATLAB.  Most of the many required constants in the model are known or determinable utilizing known constants.  The rate of change of the number concentrations of oil droplets at time t = 0 (Table 1) was very different from what we expected.  Before assuming that anything about the model was incorrect, I meticulously checked my code and calculations for correctness. 


Table 1: Rate of Change at Time t = 0

Droplet Diameter,


Number Concentration, Ni

Rate of

Change in Ni, dNi/dt





5.92 •1012

-3.79 •1010


4.00 •1012

-2.44 •1010


1.93 •1012

-1.40 •1010


5.10 •1011

6.10 •109


3.35 •1011

9.37 •108


1.57 •1011

-5.72 •108


4.89 •1010

3.74 •108


6.89 •109

1.09 •108


1.89 •109

1.98 •107


7.78 •108

1.31 •107


1.11 •108

4.51 •105



1.31 •104


1.11 •108

-7.94 •103



3.97 •103





Although the initial rate of change in the number distribution of oil droplets was much faster than anticipated, the calculations in my code appear to be correct.  Therefore, I assumed that these values were predicted by the model.  Having made this assumption, I calculated the rate of change over time.  Laboratory tests have been conducted in our lab over a period of 24 hours and we wish to study the effect of coalescence on droplet-size distribution over a period of several days. 


I utilized a one-step Runge-Kutta differential equation solver to convert the rates to number concentrations at a series of time points.  Due to the very large initial rates of change, major changes are seen in the droplet-size distribution very quickly.

Figure 2: (a) Actual laboratory data, (b) Model prediction of number concentrations


The actual laboratory data and the model prediction of the number concentrations are shown above in Figure 2 (the graphs show only the smaller droplet sizes to give a clearer view of the changes taking place).  There are visible changes in the droplet-size distribution from experimental data over the course of 24 hours.  Although the model appears to correctly predict the trends of the changes in the droplet-size distribution seen in the laboratory, it predicts that the changes seen in actual experiments should take place in only a matter of minutes.


Future Work


The Smoluchowski model does not take droplet resurfacing into account.  The largest droplets always fall into the largest volume bin and thus are never considered to resurface.  However, in laboratory experiments, the oil at the air-water interface was shown to increase with time.  Also, the discrete coalescence model implemented in MATLAB does not currently conserve volume.  These are two drawbacks that will need to be addressed in the future.




[1]        Ernest, A.N., Bonner, J.S., Autenrieth, R.L. “Determination of Particle Collision Efficiencies for Flocculent Transport

            Models,” Journal of Environmental Engineering, vol. 121, no. 4, pp. 320-239, 1995.

[2]        Levenberg, K.,"A Method for the Solution of Certain Non-linear Problems in Least-Squares," Quarterly of Applied

            Mathematics, vol. 2, no. 2, pp. 164-168, 1944.

[3]        Marquardt, D., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," SIAM Journal of Applied

[4]        Sterling, M., Bonner, J., Ernest, A., Page, C., Autenrieth, R., “Chemical Dispersant Effectiveness Testing: Influence

            of Droplet Coalescence,” Marine Pollution Bulletin, vol. 48, no. 9, pp. 969-977.