## Abstract

The release of dispersant from an aircraft onto an oil spill is simulated using the AGDISPpro computer model, to develop a better understanding of how aircraft type, spray systems, and meteorological conditions affect the prediction of surface deposition. This model, originally developed for predicting the aerial release of pesticides for agricultural spray applications, is ideally suited to simulate the effects of aircraft type and flight condition/configuration, spray system arrangement, wind speed and direction, temperature and relative humidity (evaporation), release height, and spray application rate when spraying an oil spill. Predictions of droplet trajectories from the aircraft to the surface, drop size distributions at the release height, and deposition profiles are compared to two historical datasets for the Lockheed C-130, from field studies conducted in 1982 and 1993. This article shows that model accuracy improves from R2 = 0.411 to 0.827 with the earlier data, to R2 = 0.885 to 0.968 with the later data, most probably because of a better understanding of nozzle locations in the 1993 data. Model accuracy also appears improved when the aircraft flies in an in-wind direction, a configuration strongly recommended in the available literature.

## Introduction

Oil spills at sea represent a significant environmental challenge. The application of chemical dispersant to these spills enhances the dilution of the oil into smaller droplets so that the concentration of total hydrocarbons (within a given volume of seawater) quickly falls below toxicity thresholds. Floating oil represents a hazard to coastal life and all organisms frequently reaching the sea surface (e.g., dolphins and whales), while oil droplets represent a risk to most aquatic life and the flora and fauna (benthos) found on the bottom of the sea, as oil spill droplets—when diluted—tend to deposit on the sea floor.

Aerial delivery of dispersant is sensitive to an application procedure that must not only control the location of the deposited spray material but also its droplet size when the oil spill surface is reached. Dispersant deposition levels on the water surface, off-target airborne drift of the droplets, and dispersant evaporation when traveling from aircraft release to the water surface must also be considered. For example, dispersant droplets must be large enough to penetrate the oil layer but small enough to settle at the oil-water interface, with a typical thickness between 0.1 mm and 1.0 mm [18].

The first widely recognized use of an aircraft to spray damaging insects was published in 1922 [9]. Since then, significant progress has been made in the development of spray aircraft, understanding the potential and need for the aerial application of agricultural spray materials, the nozzles that deliver the spray materials accurately and efficiently, and the spray materials themselves. Numerical techniques have been developed and applied toward a better understanding of the mechanics of the spray process and their effect on the aerial release of spray material. It is prudent to consider how this development process has impacted the use of dispersant onto oil spills.

The first published use of aircraft in the oil spill problem can be traced to a field study where spray booms were placed on two aircraft, a Piper PA-25 Pawnee and a Douglas DC-4 [10]. Later field studies evolved toward larger fixed-wing aircraft, including the Douglas Aircraft DC-3 and DC-6, Canadair CL-215, and Lockheed C-130 [4], since oil spills generally occur far from shore, time is of the essence, and the size of the dispersant payload is therefore important. Helicopters were also investigated for shorter-range applications, including the MBB Bo 105, Sikorsky S61-N, and the Aérospatiale AS 330 Puma carrying buckets typically used for fighting forest fires.

For fixed-wing applications, the aircraft wake is represented by wingtip, flap, fuselage, and tail vortices, which play an important role in droplet breakup and the deposition pattern of the released dispersant on the water surface. Later publications suggest an evolving understanding of the role played by these vortices [3,1115]. The initial development of the airborne dispersant delivery system (ADDS) and the later development of the modular aerial spray system (MASS) for the Lockheed C-130 provided two deposition datasets [3,14] spraying Corexit 9527 dispersant (Corexit Environmental Solutions LLC, Sugar Land, TX) to field study sites. These datasets provide the basis for the model application discussed herein.

## Model History

In 1979, the National Aeronautics and Space Administration supported the initial development of AGricultural DISPersal (AGDISP), a Lagrangian-based droplet trajectory prediction model for the aerial application of pesticides. This modeling technology—with funding continued to 2020 by the USDA Forest Service—was made technically feasible by previous U. S. government-funded research undertaken by Continuum Dynamics, Inc. (CDI) and others, directed at understanding the physics of vortex wakes behind large commercial aircraft, particularly on takeoff and landing, behavior close to the ground where aerial spraying (pesticides or dispersants) is conducted [16,17]. The first published version of the model [18] was followed by subsequent model improvements and extensive comparisons to field data [1921]. Additional model discussion and comparisons are reported in Refs. [2229]. The latest model improvements and comparisons to data may be found in Ref. [30]. The AGDISP/AGDISPpro model is summarized in  Appendix.

## Methods

In aerial spray operations—whether pesticide sprays or oil dispersants—the dominant mechanism driving released material toward the ground is the induced flow field generated by the aircraft wake vortices (providing that the spray is not entrained by the engine wash). Flow field models for the Lockheed C-130, in both spraying configurations (ADDS and MASS), were developed following standard accepted practices and represented by trailed wake vortices from the wingtips, the wing flap edges, the aircraft fuselage, and the horizontal tail (Fig. 1) using the weighted components identified in Table 1 [31] and the physical characteristics compiled in Table 2. A description of each flow field and development of the corresponding Corexit 9527 drop size distributions follow.

Fig. 1
Fig. 1
Close modal
Table 1

Fraction of aircraft weight carried by each component for a jet transport [31]

ComponentFraction of weight
Wing (with nacelles)0.959
Fuselage0.127
Tail−0.086
ComponentFraction of weight
Wing (with nacelles)0.959
Fuselage0.127
Tail−0.086
Table 2

Lockheed C-130 spraying weight, spraying speed [14], and aircraft parameters (Lockheed Martin C-130 brochure, 2015)

ParameterValue
Average aircraft weight63,050 kg
Spraying speed71.9 m/s
Wingspan40.40 m
Span of flaps24.16 m
Aircraft body diameter4.40 m
Span of tail16.06 m
ParameterValue
Average aircraft weight63,050 kg
Spraying speed71.9 m/s
Wingspan40.40 m
Span of flaps24.16 m
Aircraft body diameter4.40 m
Span of tail16.06 m

### Lockheed C-130 Flow Field Modeling.

Reference [14] details a field study conducted in 1982 that included two days of testing, with a Lockheed C-130 flying at four heights and spraying Corexit 9527 dispersant through Tee Jet 6140 nozzle bodies (Spraying Systems, Glendale Heights, IL), without nozzle tips. Nozzles were positioned along two spray booms that extended from the open cargo bay, generating an effective boom width of 12.5 m (without nozzles across the open bay). This spray boom and nozzle configuration appear to form an initial version of ADDS [4,13,32]. Data collection included Kromekote cards for droplet size analysis and a sectioned trough for deposition.

A later field study [3,33] describes scoping studies that tested several methods for characterizing the swath width of Corexit 9527 sprayed from fixed-wing aircraft [34]. An extensive three-day set of field tests [3] examined the usage of the Lockheed C-130 and other aircraft. In these applications, Corexit 9527 was released through an initial version of MASS. This system included spray booms positioned below the trailing edges of the wings, near their tips, and spray booms extended through holes in the parachute doors [35]. Data collection included Mylar cards, monofilament and cotton strings, oil sensitive paper for droplet size analysis, metal trays, and a sectioned trough for deposition.

Point vortex theory [36] was used to track the behavior of the aircraft wake generated by the mutually induced movement of four vortex pairs (on the left and right sides of the aircraft) and their image vortices (used to simulate the surface) and were implemented into a modified version of AGDISPpro. Results—following the paths of the vortex centerlines from initiation to decay of the vortices—are shown in Figs. 2 and 3 for the two datasets. The 42% higher spraying speed in the 1993 data (Fig. 3) lowers the tip, flap, and tail vortex strengths equally, when compared to the 1982 data (Fig. 2) but increases the fuselage vortex strength. This change modifies the mutual vortical impact on the velocity field into which the dispersant was released.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

The bluff-body flow effect induced at the rear of the cargo bay of the Lockheed C-130 (open in 1982, closed in 1993) generates additional vortical motion directly behind the aircraft [3740]. Numerical simulations [41] computed an additional vertical velocity effect for over two chord lengths downstream of the cargo bay door, open or closed, confirmed with separate calculations [42] computing a consistent centerline vertical velocity greater than 9.0 m/s within the same distance downstream of the open bay door. This additional effect was assumed to increase the fuselage vortex strength and was included in the calculations discussed below.

### Lockheed C-130 Drop Size Distributions.

Simulating the release of spray material from the Lockheed C-130 requires an estimate of the initial drop size distribution (not measured by [14] or [3]) generated by the nozzles. Since the tips were removed from the Tee Jet 6140 nozzles in the 1982 test, it was assumed here that the nozzles acted as small-diameter tubes with a known cross-sectional area of 0.495 cm2 [14]. To estimate this distribution, data on the effects of jet fuel jettisoned through exit ports on U. S. Air Force aircraft [43] were applied here. This study examined drop size distributions of water (as a substitute for jet fuel) at flow speeds comparable to dispersant discharge from the Lockheed C-130. Laser instrument readings (with a Sympatec LA-HelosTM) were interpreted for the droplet diameters Dv0.1, Dv0.5, and Dv0.9 (µm), diameters below which droplets constitute 10%, 50%, and 90% of the total volume, respectively. Dimensional analysis [4446] was applied to each of the three droplet diameters, scaled with the relative velocity between the tunnel air velocity Uair and the water jet velocity Ujet in m/s and nozzle area A (cm2) based on the equation
$D=a(Uair−Ujet)bAc$
(1)
where constants a, b, and c were found by least squares analysis (Table 3). Note that the power laws on relative velocity (constant b) averaged -2.305, consistent with the power law on Weber number scaling [47], where, for a specific value of Weber number, droplet diameter is inversely proportional to the square of the relative velocity.
Table 3

Modeling constants developed from wind tunnel test data

Droplet Size (µm)a [µm (m/s)b/cm2c]bc
Dv0.11.373 × 106−2.4980.228
Dv0.58.873 × 106−2.4130.150
Dv0.93.818 × 106−2.0050.125
Droplet Size (µm)a [µm (m/s)b/cm2c]bc
Dv0.11.373 × 106−2.4980.228
Dv0.58.873 × 106−2.4130.150
Dv0.93.818 × 106−2.0050.125

Kromekote cards measured droplet sizes on the surface, while a sectioned deposition trough recovered deposition levels across the assumed swath width of 38 m. An average of the 22 trials found UairUjet = 67.5 m/s. The field tests applied Corexit 9527. A NALCO Product Sheet (NALCO Environmental Solutions, Sugarland, TX, 2016) lists the evaporation rate of Corexit 9527 as 0.1 (water was referenced as 1.0). Since the dispersant was applied neat [4], predictions assumed that 10% of the released spray was volatile and used the quadratic evaporation rate of water [48] as a substitute for the evaporation rate of Corexit 9527.

The goal of the flight tests was to deposit Dv0.5 values between 350 µm and 500 µm. The average AGDISPpro prediction of Dv0.5 at the surface was 457.5 µm (Table 4). Model predictions recovered droplet diameters on the surface for dispersant released at three heights (9.1, 15.2, and 30.5 m) based on meteorological conditions (Table 5) estimated from historical records for the test location (Chandler, AZ) and dates (Nov. 3 and 4, 1982), as detailed meteorological data were not recorded by [14]. A logarithmic velocity profile was assumed for the wind speed between the dispersant release height and the ground.

Fig. 4
Fig. 4
Close modal
Table 4

Computed droplet size comparisons for the Lockheed C-130 (1982 data). Equation (1) was used to develop the “Initial” droplet sizes summarized in the first row of the table. The “Final” droplet sizes average all AGDISPpro model predictions for the 1982 data

DistributionDv0.1 (µm)Dv0.5 (µm)Dv0.9 (µm)
Initial (average at the nozzle exit)31.5307.7751.5
Final (average of release heights)128.5457.5778.2
DistributionDv0.1 (µm)Dv0.5 (µm)Dv0.9 (µm)
Initial (average at the nozzle exit)31.5307.7751.5
Final (average of release heights)128.5457.5778.2
Table 5

Meteorological conditions estimated during the Lockheed C-130 spray trials outside Chandler, AZ (1982 data and www.almanac.com) and Alpine, TX (1993 data and www.almanac.com)

ParameterData (11/03/82)Data (11/04/82)Data (04/27/93)
Wind speed3.50 m/s1.29 m/s3.86 m/s
Temperature18.6 °C17.4 °C18.4 °C
Relative humidity15.9%17.9%45.0%
Atmospheric pressure1020 mb1018 mb1013 mb
Wet bulb temperature depression10.83 °C10.06 °C6.61 °C
ParameterData (11/03/82)Data (11/04/82)Data (04/27/93)
Wind speed3.50 m/s1.29 m/s3.86 m/s
Temperature18.6 °C17.4 °C18.4 °C
Relative humidity15.9%17.9%45.0%
Atmospheric pressure1020 mb1018 mb1013 mb
Wet bulb temperature depression10.83 °C10.06 °C6.61 °C

Equation (1) was also applied to the 1993 data, with the averages finding UairUjet between 79.3 m/s and 92.0 m/s for the flights of interest. The initial droplet sizes for Dv0.1, Dv0.5, and Dv0.9 are shown in Table 6 (these droplet sizes are noticeably smaller than for the 1982 field studies). The goal of the 1993 flight tests was to have Dv0.5 values between 350 µm and 700 µm deposited on the surface [3]. Oil sensitive paper was used to measure droplet sizes.

Table 6

Computed droplet size comparisons for the Lockheed C-130 (1993 data)

DistributionDv0.1 (µm)Dv0.5 (µm)Dv0.9 (µm)
Initial (average at nozzle exits)17.1170.4459.4
Final (average on surface)80.8266.9478.8
DistributionDv0.1 (µm)Dv0.5 (µm)Dv0.9 (µm)
Initial (average at nozzle exits)17.1170.4459.4
Final (average on surface)80.8266.9478.8

Meteorological conditions (Table 5) were estimated from historical records for the test location and date, supplemented by additional data from Ref. [3] and the assumption of a logarithmic velocity profile.

The model predictions shown here include the effects of evaporation, removing on average 7.3% of the released spray for the eight Lockheed C-130 flights. Evaporation removes droplet sizes across the original drop size distributions as the droplets descend to the surface.

## Results and Discussion

Model predictions for both spraying configurations (ADDS and MASS) from data provided in Refs. [3,14], along with historical almanac meteorological data where necessary, are now described.

Four ADDS deposition flights were plotted by Ref. [14]. This dataset raised the following modeling issues that required addressing:

• Meteorological conditions at the time of each flight were not reported, so almanac temperature and relative humidity values were assumed for evaporation effects. The absence of wind speed information required an assumption of the value of wind speed during each flight. A systematic evaluation was performed to iterate for the wind speeds that best located the horizontal position of the peak depositions recorded in each of the four flights, as shown in the model comparisons in Figs. 47. Photographs dated from 1982 clearly show dispersant streams released from each side of the aircraft by ADDS which, based on vortical effects, appear to combine into a single deposition peak (in Figs. 4, 5, and 7) and remain two peaks (in Fig. 6) by the time the ground was reached [4,49].

• Deposition was measured across a swath width of 38 m. Any deposition occurring outside this width was artificially added by Ref. [14] to the measured deposition by an “extended numerical approximation” to recover 100% of the released dispersant on the ground. With uncertainties present in this unexplained approach, only measured data within the swath are shown in the model comparisons in Figs. 47. Integrating the measured deposition data, multiplying by the aircraft speed (71.9 m/s), and dividing by the Corexit 9527 flowrates tabulated by [14] recovered the percentage of spray that was deposited within the 38 m swath and shown in these figures.

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

Predicted peak deposition levels correlate strongly with the data, but not when filling in the deposition on either side of the peaks. As will be shown through analysis of the second Lockheed C-130 dataset, these discrepancies are most likely due to assumptions regarding the locations of the spray boom and nozzles (approximated from available historical documents), and the complexities of the unsteady bluff-body flow field generated by the open cargo bay door. The average coefficient of determination between the field data and model deposition predictions was R2 = 0.648. Table 7 summarizes model predictions.

Table 7

Summary of model predictions for the 1982 data [14]. Tabulated values are referenced to the released dispersant as 100%. Deposition levels within the 38 m swath are shown in the second column; the other three columns report the complete model deposition predictions, the predicted drift downwind of the sectioned trough, and the corresponding evaporation levels

FlightPredicted trough deposition (%)Predicted total deposition (%)Predicted total drift (%)Predicted total evaporation (%)
A-374.094.10.05.9
B-649.183.59.66.9
B-968.884.110.45.5
B-1051.881.712.36.0
FlightPredicted trough deposition (%)Predicted total deposition (%)Predicted total drift (%)Predicted total evaporation (%)
A-374.094.10.05.9
B-649.183.59.66.9
B-968.884.110.45.5
B-1051.881.712.36.0

### Lockheed C-130 With MASS.

MASS deposition flights were grouped into four plots, each plot averaging the deposition measured from three flights, with data aligned with the maximum deposition level by [3]. As with the ADDS dataset, the 1993 dataset raised several modeling issues that required addressing:

• Meteorological conditions for 5 of the 12 flights were not measured; thus, predictions for these five flights assumed the corresponding almanac temperature and relative humidity values for evaporation effects. Since three flights were averaged for each plot, it made sense to iterate for the wind speeds that best located the horizontal position of the peak depositions recorded, as shown in the model predictions compared to data in Figs. 811.

• Integrating the measured deposition data, multiplying by the aircraft speed (102.4 m/s), and dividing by the Corexit 9527 flowrates tabulated by [3] recovered the percentage of spray that deposited within the trough and shown in these figures.

• Nozzle diameters were not reported in Ref. [3] and were assumed to be the same as in the 1982 flight tests.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

These plots show significantly better agreement between model predictions and measured data (model predictions are summarized in Table 8). Note that the predicted evaporation levels are higher than in the 1982 dataset due to the higher release height in the 1993 dataset. The average coefficient of determination between the field data and model deposition predictions was R2 = 0.938.

Table 8

Summary of model predictions for the 1993 data [3]. Tabulated values are referenced to the released dispersant as 100%. Deposition levels across the sectioned trough (91.4 m) are shown in the second column; the other three columns report the complete model deposition predictions, the predicted drift downwind of the trough, and the corresponding evaporation levels

FlightsPredicted trough deposition (%)Predicted total deposition (%)Predicted total drift (%)Predicted total evaporation (%)
16, 17, 1880.585.35.88.9
19, 20, 2181.288.33.48.3
22, 23, 2468.676.514.59.0
25, 26, 2771.380.311.97.8
FlightsPredicted trough deposition (%)Predicted total deposition (%)Predicted total drift (%)Predicted total evaporation (%)
16, 17, 1880.585.35.88.9
19, 20, 2181.288.33.48.3
22, 23, 2468.676.514.59.0
25, 26, 2771.380.311.97.8

### MASS Accountancy.

Aerial application field tests and their modeling require an accounting of the destination of the spray material released from the aircraft, whether it be a water-based spray or Corexit 9527. Total accountancy and the environmental fate of these materials include predicting the evaporated vapor, ground and canopy deposition, and material aloft beyond the test area [50]. Tracking the evaporated vapor (up to 10% of the released dispersant in the case of Corexit 9527) accounts for the decrease in droplet sizes throughout the calculation, canopy deposition accounts for the removal of released dispersant by plant life, ground deposition (representing the deposit on ground collectors) recovers the dispersant pattern, and material aloft estimates what amount of dispersant is aloft beyond the ground collectors. This accountancy is summarized for the two Lockheed C-130 tests in Table 9. The higher MASS release height, compared to the average ADDS release height, results in an expected increase in evaporation, decrease in deposition, and increase in downwind drift. All calculations assumed a specific gravity of 1.0. Predictions not shown here suggest that small variations in specific gravity for Corexit 9527 did not affect the predicted results for either the ADDS or MASS systems.

Table 9

Summary of average model predictions for Corexit 9527 spray released from the Lockheed C-130 (Figs. 47 for ADDS and Figs. 811 for MASS). There is little difference between the ADDS and MASS values, even though the average release height for ADDS was 17.5 m, compared to the release height for MASS of 30.5 m

Spray despositionADDS values (%)MASS values (%)
Evaporation6.18.5
Deposition85.882.6
Downwind drift8.18.9
Spray despositionADDS values (%)MASS values (%)
Evaporation6.18.5
Deposition85.882.6
Downwind drift8.18.9

## Conclusions

Deposition results from spraying Corexit 9527 dispersant from Lockheed C-130 aircraft have been predicted with the AGDISPpro computer model. Predictions consistent with field measurements demonstrate the impact of crosswind effects on the peak and spread of the deposited material. AGDISPpro provides a physics-based model to enable a better understanding of the application of dispersant to oil spills. The behavior of dispersant—once it reaches the surface—may alter the average deposition level achieved by spraying, especially if the dispersant spreads rapidly from its predicted peak value to coat the oil spill surface at a more uniform deposition level.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

### Appendix: AGDISPpro Model Equations

AGDISPpro tracks the motion of spray droplets released from nozzles positioned on a spray boom, with one droplet released from the center of each nozzle for each droplet size in the discretized drop size distribution. The Lagrangian approach followed here partitions the variables into mean and fluctuating components (Xi + xi for droplet location (m), Vi + vi for droplet velocity (m/s), and Ui + ui for background velocity (m/s), where the indices are not summed) to give the equations
$d2dt2(Xi+xi)=[(Ui+ui)−(Vi+vi)][1τp]+gi$
(A1)
$ddt(Xi+xi)=(Vi+vi)$
(A2)
where t is time (s), Xi is the mean location of the droplet (m), xi is the fluctuating location of the droplet (m), Vi is the mean velocity of the droplet (m/s), vi is the fluctuating velocity of the droplet (m/s), Ui is the mean background velocity (m/s), ui is the fluctuating background velocity (m/s), gi is gravity (0, 0, −g) (m/s), and τp is the droplet relaxation time (s)
$τp=43ρDCDρa|Ui−Vi|$
(A3)
where ρ is the droplet density (kg/m3), D is the droplet diameter (µm), CD is the droplet drag coefficient (non-dimensional), and ρa is the air density (kg/m3). Equations governing the mean transport of a released droplet may then be written by ensemble averaging Eqs. (A1) and (A2)
$d2Xidt2=[Ui−Vi][1τp]+gi$
(A4)
$dXidt=Vi$
(A5)
The drag coefficient CD in Eq. (A3) is evaluated empirically for spherical droplets [51] as
$CD=24Re[1+0.197Re0.63+0.00026Re1.38]$
(A6)
where the Reynolds number (non-dimensional) is defined as
$Re=ρaD|Ui−Vi|μa$
(A7)
and µa is the viscosity of air (kg/m/s).
The fluctuation equations are obtained by subtracting Eqs. (A4) and (A5) from Eqs. (A1) and (A2), respectively, pre-multiplying appropriately by xi and vi, ensemble averaging and manipulating, to yield the equations
$ddt⟨xixi⟩=2⟨xivi⟩$
(A8)
$ddt⟨xivi⟩=[⟨xiui⟩−⟨xivi⟩][1τp]+⟨vivi⟩$
(A9)
$ddt⟨vivi⟩=2[⟨uivi⟩−⟨vivi⟩][1τp]$
(A10)

These correlations represent $⟨$xixi$⟩$ as the position variance (m2) around the mean droplet location Xi, $⟨$xivi$⟩$ as the correlation between droplet location and velocity (m2/s), and $⟨$vivi$⟩$ as the droplet velocity variance (m2/s2), requiring the specification of $⟨$xiui$⟩$ and $⟨$uivi$⟩$, in Eqs. (A9) and (A10), correlations of the droplet location and velocity with the local background velocity, respectively, before solution is possible. The lengthy derivation for these correlations is not reproduced here but may be found in Ref. [19].

The droplet evaporation model was originally based on the diameter-squared law [52] in which the time rate of change of a droplet diameter is approximated by
$dDdt=−D2τe[1−tτe]$
(A11)
where τe is the evaporation time scale of the droplet (s)
$τe=2DO2λΔΘSh$
(A12)
DO is the initial droplet diameter (µm), λ is the evaporation rate (µm2/s/°C), ΔΘ is the wet bulb temperature depression (°C), and Sh is the Sherwood number, equal to 2(1 + 0.27Re0.5). The evaporation rate is obtained by laboratory experiment and was initially set to 84.76 µm2/s/°C [53]. The wet bulb temperature depression is evaluated from the Carrier equation [54] and the saturation line in the steam tables [55]. Equation (A11) can be integrated to give
$1−D2DO2=tτe$
(A13)
The evaporation model described by Eqs. (A12) and (A13) represents the evaporation of isolated droplets and assumes that every droplet in the released spray responds individually to ambient meteorological conditions, even though the spray nozzles eject streams of multiple droplets with multiple droplet sizes. The physics of the spray evaporation problem suggests that spray cloud effects play a significant role in evaporation. To this end recent laboratory tests idealized the effect of a spray cloud by stacking droplets on multiple threads, each thread positioned farther downwind of the thread ahead of it [48,56,57]. This thread configuration captures the near-neighbor droplet effects on droplet evaporation and acts as an indicator of the potential change in droplet evaporation when droplets form a spray cloud. In this way, the isolated evaporation rate is measured on droplets on the most upwind threads, and spray cloud effects on evaporation are measured on the most downwind threads (on droplets experiencing evaporation rates consistent with the evaporation of spray material released within a mini cloud of droplets from a spray nozzle). Results from this study extend the isolated droplet evaporation model to achieve a closer simulation of cloud effects, providing sufficient data to reach two conclusions: (1) a more appropriate representation of evaporation behavior for droplets inside a spray cloud is one that is quadratic in time; and (2) as the Reynolds number of the droplet decreases toward zero, the evaporation rate appears to decrease to one-half its isolated droplet value. These laboratory experiments suggested a more appropriate droplet evaporation model, replacing Eq. (A13) with
$1−D2DO2=atτe[1+btτe]$
(A14)
with the parameters a = 0.2228 and b = 0.3136 (R2 = 0.959). The evaporation rate, for Re < 5, can be expressed as
$λλO=0.5+0.27812Re−0.051249Re2+0.0031249Re3$
(A15)
where λO is the evaporation rate at Re > 5 (with R2 = 0.554), now set to 67.33 µm2/s/°C [57].

The Lagrangian trajectory model in AGDISPpro solves for the behavior of each droplet in the simulation, beginning at the nozzles on the spray boom and ending when the droplet either hits the ground or moves past the downwind edge of the assumed solution space. The equations described above are solved exactly with a sufficiently small time-step.

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