Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant Cuticles: Lipophilic Surfactant Effects
Abstract
:1. Introduction
2. Model Framework
Numerical Solution Procedure
3. Results and Discussion
Sensitivity Analysis
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Additional Droplet Evaporation Equations
References
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Parameter | Definition | Value and Units | Comments |
---|---|---|---|
Control volume area | m | ||
Drop surface contact area | m | Surface contact area of drop on cuticle surface | |
Initial drop surface contact area | m | Surface contact area of drop on cuticle surface, [58] | |
AI | Active ingredient | ||
b | Thickness of cuticle | m | [59] |
Concentration of AI in drop at | mol/m | [22] | |
Concentration of adjuvant in drop at | 1 mol/m (1 g/L) | ||
Pure water concentration at and | 55,409.78 mol/m | calculated | |
Point of deliquescence concentration | mol/m | [13] | |
CCR | Constant contact radius evaporation mode | ||
CCA | Constant contact angle evaporation mode | ||
Concentration of component i | mol/m | ||
Self/bulk diffusion coefficient of AI | 7.93 ms | For CaCl, Ca diffuses the slowest, so Ca value is used, [60] | |
Self/bulk diffusion coefficient of water | 2.299 ms | [61] | |
Self/bulk diffusion coefficient of adjuvant | 7.93 ms | (fitted) | |
Diffusivity of water in air | 2.4 ms | [62] | |
Diffusivity of component i | ms | [63] | |
Fractal scaling dimension | 1.15 (-) | (fitted) | |
Functional variation of | [64] | ||
Functional of | [64,65] | ||
H | Relative humidity | 0.7 (70%) | [22] |
HLB | Hydrophilic Lipophilic Balance | ||
i | Component AI (CaCl), or ADJ (RSO 5) | ||
Adsorption rate constant ionic AI | 4.2 m(s mol) | (fitted) | |
Desorption rate constant ionic AI | 6.5 m(s mol) | (fitted) | |
Adsorption rate constant adjuvant | 2 m(s mol) | (fitted) | |
Desorption rate constant adjuvant | 5 m | (fitted) | |
Kinetic rate constant ionic AI | 5.9 m | (fitted) | |
Kinetic rate constant adjuvant | 8 m | (fitted) | |
Kinetic rate constant ionic AI | mol | calculated | |
Kinetic rate constant adjuvant | mol | calculated | |
L | Control volume length | 1 m | |
Molecular weight of CaCl2 | 110.98 g/mol | ||
Molecular weight of RSO 5 | 992 g/mol | [66] | |
Molecular weight HO | g/mol | ||
Equilibrium mass of water absorbed per CaCl2 applied | gg | [13] | |
Avogadro constant | mol | ||
Number of aqueous pores on area (1 m) of cuticle | (-) | ||
Saturated water vapour pressure in air at | 2338.8 Pa | [67] | |
RH for CaCl2 [68] and RH for CaCl2 with RSO 5 | |||
R | Gas constant | 8.3145 PamK/mol | |
RSO | Rapeseed oil surfactant | ||
Initial drop contact radius | m | Contact radius of drop on cuticle surface [58] | |
Van der Waals radius of a water molecule | m | [16] | |
Maximum radius of aqueous pores | m | For tomato fruit cuticle, ([69] p. 87) | |
Relative humidity in text | |||
t | Time | s | |
T | Temperature | 293.15 K () | [22] |
u | Bound integration variable | ||
Volume of droplet at | 1 m | [22] | |
Volume of water in droplet at time t | |||
Deliquescent droplet volume | m | ||
Partial molar volume CaCl | mmol | [70] | |
Partial molar volume adjuvant | mmol | ||
Partial molar volume water | mmol | [71] | |
x | Length | m | |
Aqueous pore pathway porosity | |||
Lipophilic pathway porosity | 0.03 | (fitted) | |
Concentration adsorbed ionic AI per droplet area on cuticle surface | mol/m | ||
Concentration adsorbed adjuvant per droplet area on cuticle surface | mol/m | ||
Saturated adsorbed molecules per droplet area | 400 mol/m | (fitted) | |
Density of aqueous pores in cuticle | m | [13] | |
Evaporation constants as a function of relative humidity | ms | ||
Saturated water vapour concentration as a function of relative humidity | g/m | [72] | |
Point of deliquescence humidity shifting factor for choosing the humidity in and | |||
Liquid density HO at | g/m | [73] | |
Liquid density of CaCl2 at | g/m | [74] | |
Contact angle of drop on cuticle surface that changes with time | rads | ||
Contact angle of drop on cuticle surface at | rads | See Table 1 in [13] | |
Receding contact angle of drop on cuticle surface | rads | See Table 1 in [13] | |
Logistic decay evaporation constant | 0.0428 Lg | [13] | |
Logistic decay evaporation term (a constant) as a function of initial concentration of ionic AI | calculated | ||
Point of deliquescence humidity shifting factor to incorporate with the addition of adjuvants | RH | [13] |
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Tredenick, E.C.; Farrell, T.W.; Forster, W.A. Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant Cuticles: Lipophilic Surfactant Effects. Plants 2019, 8, 202. https://doi.org/10.3390/plants8070202
Tredenick EC, Farrell TW, Forster WA. Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant Cuticles: Lipophilic Surfactant Effects. Plants. 2019; 8(7):202. https://doi.org/10.3390/plants8070202
Chicago/Turabian StyleTredenick, Eloise C., Troy W. Farrell, and W. Alison Forster. 2019. "Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant Cuticles: Lipophilic Surfactant Effects" Plants 8, no. 7: 202. https://doi.org/10.3390/plants8070202