Coalescence During Chemical Dispersion of Oil Spills Amanda Virkus Supervised by Dr.
Brian Wrenn Background |
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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. Coalescence |
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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]: (1) where ∂N_{k}_{ }/∂t is the rate of change in the number of droplets of volume v_{k} due to coalescence, α represents the collision/sticking efficiency (the fraction of effective collisions), β(v_{i} , v_{j}) represents the frequency of collisions between droplets of volume v_{i} and v_{j} (where v_{i} + v_{j} = v_{k}), and n_{i} is the number of droplets of volume v_{i}. The first term represents the formation of droplets of size v_{k} from the coalescence of smaller droplets. The second term represents the loss of droplets of size v_{k} from the coalescence of droplets of size v_{k} 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(v_{i} , v_{j}), 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. (2) where frequency of collisions due to turbulent shear can be determined using the following model [1]: (3) 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 |
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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: (5) 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.
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 |
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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. References |
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[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,"
[4]
of
Droplet Coalescence, Marine Pollution Bulletin, vol. 48, no. 9, pp.
969-977. |