# A Fast Analysis of Pesticide Spray Dispersion by an Agricultural Aircraft Very near the Ground

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Terminology

**x**be a point in the three-dimensional space with the coordinates (x,y,z), and the fluid velocity

**u**in terms of time t and space point

**x**is denoted as

**u**(

**x**,t) = (u,v,w). The critical Mach number for a low speed aircraft is 0.3, and when Ma < 0.3, the compression effect is neglected and is treated as incompressible flow. The cruise speed of an agricultural spraying monoplane is typically less than 100 m/s, and it belongs to a category of low-speed aircrafts.

**ω**is known, the flow field velocity

**u**is induced by

**ω**and can be calculated according to the Biot–Savart law:

_{∞}equivalent to flight speed; the included angle between freestream and the chord line c(y) is the angle of attack α(y).

#### 2.2. Induced Velocity Field

#### 2.2.1. Lifting Line Model

_{L =}

_{0}be the airfoil angle of attack when lift L is 0 and Γ(y) be the circulation developed over the airfoil at cross section y. According to the lifting line model, there is the following equation [12]:

_{∞}, α

_{L =}

_{0}are all known parameters about airfoil geometry and flight conditions. The function in (2) to be solved is the airfoil circulation distribution Γ(y). The above equation is a singular integral equation and has been very well investigated in mathematical theory and numerical analysis [24].

**u**= (u,v,w) at point A is:

**u**for each point in space. Equation (2) and Formula (4) generally do not have an analytical solution, but their numerical solutions can be completed in a very short time, typically 2.1 s on a common PC (CPU 2 GHz, memory 2 GB). The related parameters in the Equation (2) have been determined by the wing structure and airfoil geometry, with the exception of flight speed V

_{∞}to which the circulation is linearly proportional. Therefore, the shape of circulation Γ(y) is directly related to the aircraft structure itself and can be acquired entirely in advance. In fact, most agricultural fixed-wing aircraft have three types of lift distribution [8], namely rectangular, elliptical, and triangular, without the need to solve Equation (2).

#### 2.2.2. Wingtip Vortices Model

_{∞}:

_{∞}is the atmospheric density. From Formula (1) it can be deduced that the tangential induced velocity by a point vortex is:

_{c}is the vortex core radius defined at which the velocity v is a maximum. This vortex core radius is in general selected to be r

_{c}= 0.052b

_{0}, b

_{0}is the vortex separation [27].

_{0},y,z) corresponding to the downstream distance x

_{0}on a plane at which a point vortex pair locates, the velocity is contributed by this certain vortex pair. The velocity is:

_{1}and r

_{2}are the distances between point P and two wingtip vortices, respectively. When the downstream distance x

_{0}varies, the value of circulation Γ changes with it because of atmospheric viscosity. Defining the initial circulation Γ

_{0}, the circulation in terms of decay time t and radial distance r is:

^{−5}kg/(m·s) in the aerial application environment. Then it is found that the changing of Γ becomes apparent only after the time t is approaching 10

^{4}~10

^{5}s, well beyond the far-field investigated in this paper. Therefore, on addressing wake vortices decay, the atmospheric turbulence is here primarily considered just ignoring the effect of air viscosity. Set q root mean square (rms) of turbulent velocity, and it has [28]:

_{0},y,z): 1. calculate the initial circulation according to Equation (6); 2. calculate the circulation at the distance of downstream x

_{0}from Formula (8); 3. determine the distances r

_{1}and r

_{2}and compute the induced velocity at point P through Formula (7); 4. decompose the tangential velocity into components of y and z direction. It should be noted that the wingtip vortices are also in motion by their self-induced velocity. The problem of position of vortex pair is to be dealt with later.

#### 2.2.3. Mixture Model

_{0}= b

^{2}/2Γ

_{0}[13]. Beyond a distance corresponding to the moment t

_{0}, it is treated as far-field where the wingtip vortices model works, otherwise it is near-field in which is the sphere of application of the lifting line model. Therefore, the lifting line-wingtip vortices mixture model is established to just match two evolving phases of trailing vortices and bypass the assumption of the parallel vortex sheet.

#### 2.2.4. Trajectory Approximation

_{p}is the particle relaxation time, g the standard acceleration due to gravity. The relaxation time τ

_{p}is computed from:

_{∞}, ρ

_{p}is the density of air and the particle. respectively, D

_{p}the particle diameter, V

_{rel}the relative velocity to the flow field. The drag coefficient C

_{D}is given by [29]:

^{−3}< Re < 2 × 10

^{5}, the Reynolds number of a particle Re is defined as:

_{p}and related aerodynamic drag are often nonlinear in terms of the relative velocity due to the particle size spread out across a wide range. It will make trajectory of tacking for an ensemble of particles computationally intensive. In order to achieve a simplified analysis, only three kinds of particles, namely very coarse, median, and very fine, are dealt with. Especially for very fine particles (smaller than 20 μm), it has a sufficiently small Reynolds number to be in the Stokes flow regime, and Equation (9) reduces to a linear form as relaxation time becomes:

_{i}and standard deviation σ

_{i}(i = 1,…, N). Then the overall population, namely the total fraction, meets with the Gaussian mixture model expressed as:

_{i}and standard deviation σ

_{i}(i = 1,…, N) is determined by positions of ground deposits of three typical particles mentioned in the above. Finally, the particle deposition spreading over the ground is resolved through the Lagrangian dynamics on considering certainty contained and the Gaussian mixture model dealing with stochasticity corresponding to the turbulent effect of the airflow and other factors not included in the deterministic strategy, such as particles with sizes out of our formulas.

#### 2.3. Wake Vortices Motion

_{0}. The primary vortices firstly sink due to their interference at a constant velocity out-of-ground effect (OGE). As they approach the ground, ground effect takes place and leads to lateral separation of them where it is labeled as the near-ground effect (NGE) region. The vortices then go further down and begin to interact strongly with the ground, entering the region of in-ground effect (IGE). The boundary layer close to ground is produced, detached, and two secondary vortices with opposite sign vorticity are generated beneath each primary vortex. The interaction of primary and secondary vortices leads the vortex rebound, rising again to a new height. Subsequently, the vortex system even induces more secondary vortices if given enough circulation, disperses, and finally is fading away in the atmosphere.

- In the OGE phase (Figure 3a), it is a two-vortex system, circulation decay subjecting to Formula (8), and the downward velocity is Γ/2πb
_{0}. - In the NGE phase, at a height of h
_{1}= b_{0}*ZIMFAC above the ground, it is a four-vortex system (Figure 3b). Two image vortices are added below the ground as the mirror of primary vortices to meet the boundary condition of zero vertical velocity at the ground. Because of the symmetry, the trajectory of only starboard vortex is addressed. Here ZIMFAC stands for “z image factor” determined by experience. It has:$$\{\begin{array}{l}\frac{dy}{dt}=\frac{\mathsf{\Gamma}}{4\pi}(\frac{1}{z}-\frac{z}{{y}^{2}+{z}^{2}})\hfill \\ \frac{dz}{dt}=-\frac{\mathsf{\Gamma}}{4\pi}(\frac{1}{y}-\frac{y}{{y}^{2}+{z}^{2}})\hfill \end{array}$$_{0}/2 expressed by:$$\frac{{d}_{0}}{2}=\frac{{b}_{0}}{\sqrt{{({b}_{0}/{h}_{0})}^{2}+4}}$$_{0}is the initial altitude that a primary vortex pair generates. - In the IGE phase, at a height of h
_{2}= b_{0}*ZGEFAC (for “z ground effect factor”) above the ground, it is an eight-vortex system (Figure 3c). The two secondary vortices and their images are introduced at a distance of b_{1}and at an initial rotation angle θ of outboard of primary vortices, where θ is zero below the primary vortices and is positive clockwise for the port vortex or counterclockwise for the starboard vortex [33]. The initial ratio between secondary and primary circulation is defined as γ. Set (y_{i},z_{i}) (i = 1,…,8) the position of vortices, this eight-vortex system is subjected to the following Equation (15) of point vortex dynamics:$$\{\begin{array}{l}\frac{d{y}_{i}}{dt}=-\frac{1}{2\pi}{\displaystyle \sum _{i\ne j}{\mathsf{\Gamma}}_{j}}\left({z}_{j}-{z}_{i}\right)/{r}_{ij}^{2}\hfill \\ \frac{d{z}_{i}}{dt}=\frac{1}{2\pi}{\displaystyle \sum _{i\ne j}{\mathsf{\Gamma}}_{j}}\left({y}_{j}-{y}_{i}\right)/{r}_{ij}^{2}\hfill \end{array}$$

_{*}is the friction velocity, κ = 0.4 is the von Karman constant, d the displacement height typically equals to 75% of the canopy height h

_{c}, z

_{0}= h

_{c}/30 indicated by laboratory measurements is the roughness length. The crosswind affects the lateral drift in the spanwise direction. The crosswind term will be added to right side of the ODEs (15) with respected to dy/dt.

## 3. Results and Analysis

#### 3.1. Induced Velocity Distribution

_{0}to be dimensionless, in which W

_{0}= Γ

_{0}/2πb. As seen from Figure 4a,b the induced velocity is symmetric about the z-axis. The velocity appears negative (downwash flow) within the wingspan, but becomes positive (upwash flow) beyond it and decays to be zero. Combined with Figure 4c,d the induced maximal downward velocity corresponding to the elliptic lift is at y = 0, z ≈ −0.1b, the wing root region, while the maximal downwash of rectangular lift approaches the wingtips.

_{0}. Therefore, an agricultural aircraft with rectangular lift is preferred in aerial spraying and the nozzles should be mounted within the downwash flow region.

_{0}= 61 m

^{2}/s. To agree with the setting in [14] the center of velocity curve is set at y = 6 m, and the comparison of velocity distribution under the wing above the ground 3 m corresponding to z = −0.14b is shown in Figure 6. The crosswind blows the starboard vortex slightly higher than the port vortex, so the velocity profile is asymmetric and the downwash value near the starboard vortex is smaller compared to the port vortex, which was also demonstrated in [31]. But to our surprise, the maximal vertical velocity error is less than 0.5 m/s even out of consideration the effect of crosswind in our model.

_{d}av, E

_{u}av of both downwash and upwash read less than 0.5 m/s within 50 s from Figure 7b, and the maximal velocity errors, E

_{d}, E

_{u}, are less than 1.5 m/s at t = 2 s, then reach the peak (less than 1.7 m/s) appearing at t = 10 s the moment vortex rebound and secondary vortices begin to happen. Simultaneously, the maximal and average relative errors also have the largest values at t = 2 s and subsequently decay as seen in Figure 7c. Although this vortex evolution mechanism has not been considered, it is found that it does affect the downwash flow field very slightly, especially in the late stage of wake evolution. In any case, from the aspect of relative errors the lifting line-wingtip vortices mixture model does not work very well as the average relative error of downwash is nearly 18% and average relative error of upwash is about 10% in the whole course of wingtip vortices life cycle. But note it is the absolute error, other than relative error, that imposes negative effects on the accuracy of analysis more so this model has a satisfactory performance for rapid velocity calculation though relative erros are large.

#### 3.2. Vortex Trajectory

_{0}, Z

_{0}) = (5.65, 5) for Thrush 510G, are already in the NGE phase indicated by Figure 3b because the vortex separation b

_{0}= 2Y

_{0}, the height Z

_{0}< h

_{1}= b

_{0}*ZIMFAC where ZIMFAC = 1.5 [33]. The image vortex allows the primary vortex to acquire an outward velocity component, while in the OGE phase the primary vortices of a two-vortex system can only have a downward velocity just induced from the other primary vortex behaving like a free fall. In different phases the patterns of starboard vortex are described in Figure 8. The NGE phase is dominated by the outward velocity significantly greater than the downward component, and then the primary vortex will approach to the certain height d

_{0}/2 = 3.74 m computed from Formula (14) but not impacting the ground. This behavior is unfavorable in aerial spraying.

_{2}= 0.6b

_{0}. Sometimes the initial height of primary vortices Z

_{0}is already within the range. But the vortex system does not fall into IGE phase immediately. The secondary vortices generation is an indication that the IGE phase needs time to evolve. The vortex system have to firstly experience NGE phase, and then transfers to the IGE phase in several seconds. This time duration depends essentially on primary vortices’ release height, initial circulation and their decaying rate. If the vortex evolving starts from the OGE phase, the full stages can be observed that it begins to descend in OGE phase, and then move outward in NGE phase, afterward the altitude approaching 0.6b

_{0}it enters IGE phase. Vortex rebound is a typical phenomenon in extreme ground effect depicted in Figure 9. The secondary vortex, which has the opposite sign of circulation, gives the primary vortex a velocity upward. Under different conditions the primary vortex in IGE phase is to be in U turn motion just moving up, or loop itself as shown in Figure 9. The accurate trajectory of vortex motion is related to the setting of relevant parameters. To agree with results described in Figure 9, characteristic parameters in IGE phase are given by Table 2. But in presence of crosswind, it is believed that the crosswind shear makes a redistribution of the vorticity that contained in the boundary layer, and the high crosswind suppresses the creation of secondary vortex, leading to the weakness of vortex rebound.

#### 3.3. Validation of Fast Analysis

#### 3.3.1. No Wind

_{m}, y

_{M}of deposition on the ground, between which other droplets own the high probability of falling. Considering droplet size distribution (Figure 10), the deposition pattern of spray particles is expressed as ideal stem plots in Figure 12. There are six stems appearing in this figure corresponding to nozzles from which droplets deposit on the ground while others disperse beyond the region. Note that the 177 μm droplets occupy the largest fraction in the volume; they are represented as peaks in the figure. It is evident that the deposition of fine droplet is close to the position of 10 μm droplet, and the coarse droplet near that of 670 μm droplet.

_{M}− y

_{m}| as the variance σ, then there is a Gaussian mixture model containing six normal distributions also shown in Figure 12. Adding these Gaussian distributions obtains the deposition curve of NGE phase from all nozzles in a spray line described in Figure 13. The droplets from two nozzles at y = ±1 m are found to be highly concentrated having the maximal deposition, since the outward velocity in the central region is small and favorable to deposition. But compared to the prediction by CFD the deposition curve is too narrow, revealing the dispersal effect of NGE phase in the spanwise direction has not been adequately accounted for.

#### 3.3.2. Effect of Crosswind

_{*}= 0.13 m/s regarding the 2 m/s crosswind at the height of 4 m over the ground, and the vertical wind speed is subjected to the logarithmic law, while the headwind is assumed to be zero. The IGE phase is considered. Trajectories of wake vortices affected by the crosswind are presented in Figure 16. Because of the wind, the symmetry of vortex trajectories is broken. The primary and secondary vortices rebound is suppressed by the crosswind and is somewhat lower than in Figure 14. For persisting at least 50 s they float in the air and hardly deposit onto the ground. The trails of fine droplets will be seriously influenced by the suspending of wake vortices.

#### 3.3.3. Effect of Headwind

#### 3.3.4. Relation between Droplet Size and Drift Distance

^{2}for “fitting1” and “fitting2” are 0.88, 0.82, respectively, indicating a significant linear correlation of the size and drift. This relationship that is associated with the wind speed and nozzle placement yet can be used in a gross analysis. In some instances, for example, the case of Section 3.3.2 the linear relation from theoretic computation is hardly satisfied.

## 4. Discussion

_{0}, droplet outlet velocity v

_{0}, then droplet trajectories are compared in Figure 23 for their different sizes and initial states.

## 5. Conclusions

- The lifting line-wingtip vortices mixture model allows rapid calculation of the complete velocity field around an agricultural monoplane in 2.1 s on a common PC (2 GHz CPU, 2 GB RAM), and the whole fast analysis for estimating droplets trajectories and drift is implemented within 3.2 s. For the same case, AGDISP takes 25 s whilst CFD needs several to tens of hours.
- The lifting line-wingtip vortices mixture model is in good agreement with the experimental and CFD results for Thrush 510G aircraft. At a height over the ground of 3 m, the maximum velocity error is less than 1.5 m/s and the average error is less than 0.5 m/s in the space that is 7.6 wingspans downstream of the aircraft (corresponding to a time span of 2 s). Outside this region, the maximum velocity error does not exceed 1.7 m/s, and the error tends to decrease with distance. The N-vortex system, by adding secondary vortices and their images, can predict vortex rebound and thereafter vortex motion, roughly matching with CFD simulation. The flight very near the ground could induce stronger secondary vortices, produce additional upwash flow, and result in entrainment of particles aloft more seriously.
- The turbulent effect of airflow and other factors that make droplets disperse randomly can be handled through a probability distribution described as the Gaussian mixture model whose parameters are determined by tracking ground deposition of some droplets with typical sizes within the Lagrangian framework.
- The fast analysis does not rely on swath width input that is required in AGDISP and is usually achieved by a preliminary experiment. The performance of this method validates that it matches well with AGDISP on predicting droplet trajectories, but makes a conservative estimate to the drift compared to AGDISP and CFD simulation. The drift or dispersion is associated with droplet size, release height, nozzle distribution, and wind speed when an agricultural monoplane and the flight parameters are determined. Generally speaking, the small release height and nozzles mounted in the middle of the wingspan will contribute to the efficient deposition. But the influence of the two factors is negligible for fine droplets. The droplet size and wind speed are the leading factors. The crosswind changes the vortex trajectory and further their induced velocity field where there exists outward velocity near the ground and droplets are taken downwind far away. The headwind affecting the droplet drift only through its spanwise component may imply the control of long distance dispersion by adjustable flight line. The drift can be suppressed by applying coarse droplets against crosswind or wake vortices.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Vortex system in different phases approaching the ground: (

**a**) out-of-ground effect (OGE) phase; (

**b**) near-ground effect (NGE) phase; (

**c**) in-ground effect (IGE) phase.

**Figure 4.**Induced vertical velocity distribution under a wing with the elliptic and rectangular lift distribution: (

**a**) yz plane at x = 0 with elliptic lift; (

**b**) yz plane at x = 0 with rectangular lift; (

**c**) z = −0.1b, −0.3b, −0.5b on yz plane (x = 0) with elliptic lift; (

**d**) z = −0.1b, −0.3b, −0.5b on yz plane (x = 0) with rectangular lift.

**Figure 5.**Induced vertical velocity distribution by wingtip vortices model on yz plane at x = 2b downstream: (

**a**) z = −0.1b, −0.3b, −0.5b below wingtip vortices; (

**b**) two-dimensional velocity field on yz plane (x = 2b).

**Figure 6.**Comparison of vertical velocity at the height z = 3 m on yz plane at x = 0, “CFD” and “Field Exp” are results of computational fluid dynamics (CFD) and field measurement in [14], “theory” referring to the lifting line-wingtip vortices mixture model proposed in this paper.

**Figure 7.**Induced velocity in far-field and error between theory and CFD [14] of Thrush 510G above the ground 3 m: (

**a**) velocity profile (spanwise) over time corresponding downstream x = 7.6b, 38b, 76b, 190b; (

**b**) absolute error over time, E

_{d}: maximal error of downwash on y-axis, E

_{u}: maximal error of upwash on y-axis, E

_{d}av: average error of downwash on y-axis, E

_{u}av: average error of upwash on y-axis; (

**c**) relative error over time, RE

_{d}: maximal relative error of downwash on y-axis, RE

_{u}: maximal relative error of upwash on y-axis, RE

_{d}av: average relative error of downwash on y-axis, RE

_{u}av: average relative error of upwash on y-axis.

**Figure 11.**Path lines of flow field in NGE phase as approximations to trajectories of 10 μm droplets.

**Figure 12.**Ideal deposition from each nozzle and their dispersal estimated by Gaussian distribution.

**Figure 13.**Comparisons of ground depositions in absence of wind predicted by CFD [19], NGE and IGE model proposed in this paper.

**Figure 15.**Path lines and trajectories of droplets with different sizes from ten nozzles in the IGE model: (

**a**) path lines; (

**b**) trajectories of 10 μm droplets; (

**c**) trajectories of 137 μm droplets; (

**d**) trajectories of 670 μm droplets.

**Figure 16.**Trajectories of wake vortices in the IGE phase in presence of wind, “PM” refers to primary vortex, “SB” being the starboard vortex, “port” the port vortex, and “SD” the secondary vortex.

**Figure 17.**Comparisons of trajectories of 10 μm droplets in the IGE model and AGDISP from two views, two respective figures in the up (IGE model) and down (AGDISP) have the same scale and unit of measure: (

**a**) yz plane (rear view of an aircraft) based on IGE model; (

**b**) xz plane (right view of an aircraft) based on IGE model; (

**c**) yz plane (rear view of an aircraft) based on AGDISP; (

**d**) xz plane (right view of an aircraft) based on AGDISP.

**Figure 18.**Comparisons of trajectories of 137 μm droplets in the IGE model and AGDISP from two views. Two respective figures in the up (IGE model) and down (AGDISP) have the same scale and unit of measure: (

**a**) yz plane (rear view of an aircraft) based on IGE model; (

**b**) xz plane (right view of an aircraft) based on IGE model; (

**c**) yz plane (rear view of an aircraft) based on AGDISP; (

**d**) xz plane (right view of an aircraft) based on AGDISP.

**Figure 19.**Comparisons of trajectories of 670 μm droplets in the IGE model and AGDISP from two views, two respective figures in the up (IGE model) and down (AGDISP) have the same scale and unit of measure: (

**a**) yz plane (rear view of an aircraft) based on IGE model; (

**b**) xz plane (right view of an aircraft) based on IGE model; (

**c**) yz plane (rear view of an aircraft) based on AGDISP; (

**d**) xz plane (right view of an aircraft) based on AGDISP.

**Figure 20.**Comparison of deposition and drift in presence of crosswind predicted by CFD [19], AGDISP and theoretic method of this paper.

**Figure 21.**Spray dispersion of Thrush 510G under the effect of headwind: (

**a**) trajectories of typical droplets with size 137 μm; (

**b**) ground deposition predicted by IGE model (labeled as “theory”) is compared to sampling data (labeled as “field”) by field experiment [2].

**Figure 22.**Relation between droplet size and drift distance compared to theory and measure of field experiment, “fitting1” and “fitting2” refer to the linear regression of data obtained by measure (field experiment) and theory, respectively.

**Figure 23.**Droplet trajectories under different conditions: (

**a**) size 10 μm, release height z

_{0}= 1 m, outlet velocity v

_{0}= 20 m/s, friction velocity u

_{*}= 0 m/s; (

**b**) size 50 μm, release height z

_{0}= 2 m, outlet velocity v

_{0}= 10 m/s, friction velocity u

_{*}= 0 m/s; (

**c**) size 50 μm, release height z

_{0}= 2 m, outlet velocity v

_{0}= 10 m/s, friction velocity u

_{*}= 0.13 m/s; (

**d**) size 180 μm, release height z

_{0}= 2 m, outlet velocity v

_{0}= 10 m/s, friction velocity u

_{*}= 0.13 m/s.

Parameter | Gross Weight | Wingspan | Flight Altitude | Release Height | Flight Speed | Air Density | Wind |
---|---|---|---|---|---|---|---|

Value | 4367 kg | 14.47 m | 5 m | 4.7 m | 55 m/s | 1.29 kg/m^{3} | 4 m/s |

Description | Distance between Primary and Secondary Vortex | Initial Rotation Angle from Primary Vortex | Initial Ratio between Secondary and Primary Circulation | z Ground Effect Factor |
---|---|---|---|---|

Symbol | b_{1} | θ | γ | ZGEFAC |

Value | 0.17b_{0} | π/10 | 0.64 | 0.6 |

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## Share and Cite

**MDPI and ACS Style**

King, J.; Xue, X.; Yao, W.; Jin, Z.
A Fast Analysis of Pesticide Spray Dispersion by an Agricultural Aircraft Very near the Ground. *Agriculture* **2022**, *12*, 433.
https://doi.org/10.3390/agriculture12030433

**AMA Style**

King J, Xue X, Yao W, Jin Z.
A Fast Analysis of Pesticide Spray Dispersion by an Agricultural Aircraft Very near the Ground. *Agriculture*. 2022; 12(3):433.
https://doi.org/10.3390/agriculture12030433

**Chicago/Turabian Style**

King, Ji, Xinyu Xue, Weixiang Yao, and Zhen Jin.
2022. "A Fast Analysis of Pesticide Spray Dispersion by an Agricultural Aircraft Very near the Ground" *Agriculture* 12, no. 3: 433.
https://doi.org/10.3390/agriculture12030433