Water saving is one of the biggest issues that the world will soon have to deal with, considering the unre- lenting population growth and the not uniform global dis- tribution of fresh water sources. Agriculture alone is re- sponsible, in many countries, for some 70% of its usage and consequently finding ways to save water in agricul- ture would produce a particularly significant result in such struggle. Sprinkler irrigation is one of the most dif- fused irrigation techniques employed in agriculture. Based on these initial considerations, my lecture will de- scribe and analyze the dynamic and thermal– fluiddynamic behavior of a water droplet travelling from the nozzle outlet to the ground through dry and moist air, in function of all the variables involved in the process. The analytical model arrived at in the present paper and successively implemented by numerical means, makes many steps ahead with respect to a previous re- search by Lorenzini [1], which provided a mathematical tool for single droplet dynamics description, based on a few simplifications which are not shared by the present approach: the volume of the droplet was invariant during the flight and evaporation was instantaneous at the end of the flight and due just to the dynamic components in ac- tion; the droplet travelled in a no-wind condition. In fact the process here considered is much more complicate as the water droplet changes its volume during the whole aerial path and so the whole thermal fluiddynamic proc- ess becomes technically realistic. Moreover droplet dy- namics and evaporation is modeled as a function of the whole range of parameters which actually affect the process, such as: droplet initial diameter, droplet initial velocity, water temperature, air temperature, diffusion coefficient to water in air, air relative humidity, droplet inlet inclination, solar and environmental radiation, wind vectorial velocity. Hence the process is here not just con- sidered as a simply dynamic or as a simply thermal fluid dynamic one but as a full combination of the two, which is a novel approach in literature. The following hypotheses have been taken to solve the problem: (1) the physical system considered is a single droplet exiting from the nozzle of a sprinkler and generated ex- actly in correspondence to the nozzle outlet; (2) the forces applied to the system are weight, buoy- ancy and friction; (3) the droplet has a spherical shape all the way down [2]; (4) friction has the same direction of velocity for all the path but opposite sense; but, differently from Lorenzini [1]: (5) the volume of the droplet is variable throughout the flight due to its continuous evaporation; (6) wind is not neglected but it is considered as a vec- torial entity, potentially affecting the droplet flight in every single direction/sense as possible; (7) air humidity is a parameter study; (8) radiation (solar + environmental) is a parameter study; moreover: (9) diffusion of air in water is negligible. It is evident that hypotheses (5), (6), (7) and (8) not only add a relevant analytical complication to the nature of the present study, but also mark a significant step in the way of an entirely realistic dynamic and evaporative description of an in-flight sprinkler water droplet, with- out any simplifications in the thermal description of the analysis variables and none of the parametric effects is maximized – such as air friction in [1] – but always de- scribed in their true dynamics.

Modelling droplet evaporation in air. Challenges and perspectives / Lorenzini, Giulio. - (2014), p. 9.

### Modelling droplet evaporation in air. Challenges and perspectives

#####
*LORENZINI, Giulio*

##### 2014

#### Abstract

Water saving is one of the biggest issues that the world will soon have to deal with, considering the unre- lenting population growth and the not uniform global dis- tribution of fresh water sources. Agriculture alone is re- sponsible, in many countries, for some 70% of its usage and consequently finding ways to save water in agricul- ture would produce a particularly significant result in such struggle. Sprinkler irrigation is one of the most dif- fused irrigation techniques employed in agriculture. Based on these initial considerations, my lecture will de- scribe and analyze the dynamic and thermal– fluiddynamic behavior of a water droplet travelling from the nozzle outlet to the ground through dry and moist air, in function of all the variables involved in the process. The analytical model arrived at in the present paper and successively implemented by numerical means, makes many steps ahead with respect to a previous re- search by Lorenzini [1], which provided a mathematical tool for single droplet dynamics description, based on a few simplifications which are not shared by the present approach: the volume of the droplet was invariant during the flight and evaporation was instantaneous at the end of the flight and due just to the dynamic components in ac- tion; the droplet travelled in a no-wind condition. In fact the process here considered is much more complicate as the water droplet changes its volume during the whole aerial path and so the whole thermal fluiddynamic proc- ess becomes technically realistic. Moreover droplet dy- namics and evaporation is modeled as a function of the whole range of parameters which actually affect the process, such as: droplet initial diameter, droplet initial velocity, water temperature, air temperature, diffusion coefficient to water in air, air relative humidity, droplet inlet inclination, solar and environmental radiation, wind vectorial velocity. Hence the process is here not just con- sidered as a simply dynamic or as a simply thermal fluid dynamic one but as a full combination of the two, which is a novel approach in literature. The following hypotheses have been taken to solve the problem: (1) the physical system considered is a single droplet exiting from the nozzle of a sprinkler and generated ex- actly in correspondence to the nozzle outlet; (2) the forces applied to the system are weight, buoy- ancy and friction; (3) the droplet has a spherical shape all the way down [2]; (4) friction has the same direction of velocity for all the path but opposite sense; but, differently from Lorenzini [1]: (5) the volume of the droplet is variable throughout the flight due to its continuous evaporation; (6) wind is not neglected but it is considered as a vec- torial entity, potentially affecting the droplet flight in every single direction/sense as possible; (7) air humidity is a parameter study; (8) radiation (solar + environmental) is a parameter study; moreover: (9) diffusion of air in water is negligible. It is evident that hypotheses (5), (6), (7) and (8) not only add a relevant analytical complication to the nature of the present study, but also mark a significant step in the way of an entirely realistic dynamic and evaporative description of an in-flight sprinkler water droplet, with- out any simplifications in the thermal description of the analysis variables and none of the parametric effects is maximized – such as air friction in [1] – but always de- scribed in their true dynamics.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.