Thermal fluid dynamic modelling of a water droplet evaporating in air

International Journal of Heat and Mass Transfer 62 (2013) 323–335

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Thermal fluid dynamic modelling of a water droplet evaporating in air Giulio Lorenzini a,⇑, Onorio Saro b a b

University of Parma, Department of Industrial Engineering, Parco Area delle Scienze, 181/A, 43124 Parma, Italy University of Udine, Department of Electric Management and Mechanical Engineering, via delle Scienze, 208, 33100 Udine, Italy

a r t i c l e

i n f o

Article history: Received 23 October 2012 Received in revised form 22 February 2013 Accepted 26 February 2013 Available online 9 April 2013 Keywords: Water droplet Thermal fluid dynamic modelling Parametrical study Numerical description

a b s t r a c t Water saving is one of the biggest issues that the world will soon have to deal with, considering the unrelenting population growth and the not uniform global distribution of fresh water sources. Agriculture alone is responsible, in many countries, for some 70% of its usage and consequently finding ways to save water in agriculture would produce a particularly significant result in such struggle. Sprinkler irrigation is one of the most diffused irrigation techniques employed in agriculture. Based on these initial considerations, the present paper is aimed at describing and understanding the dynamic and thermal– fluiddynamic behaviour 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, namely: droplet initial diameter, droplet initial velocity, water temperature, air temperature, diffusion coefficient of water in air, air relative humidity, droplet inlet inclination, solar and environmental radiation, wind vectorial velocity. After a full analytical modelling of the phenomenon, a numerical implementation based on Runge–Kutta fourth order method was made. The effect of the above reported parameters is broadly discussed also in relation to their in-flight evolution; as foreseen in previous papers, also air friction proves not to be negligible when assessing in-flight droplet evaporation. Some comparisons with well established literature data contribute to the model validation. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In addition to that concerning energy, one of the big questions arising to the current generation is whether in a relatively near future there will be a chance that fresh water offer and demand can match. To prevent a negative answer a rational use and a serious global water saving policy is nowadays becoming unavoidable, especially due to the increasing world population and to a not uniform distribution of such precious and strategic resource. An effective way to reduce global water waste is to reduce agricultural water waste as, in many (even economically and technologically developed) countries, the agricultural supply chain is responsible of up to 70% of fresh water total usage ([32,37]; www.ecpa.eu/ page/water-use). This is why saving in the agricultural processes a certain percentage of water would quantitatively help reduce the waste much more than acting in any other contexts such as the industrial or the civil. In a sense this would mean water saving optimisation. With such conviction this research focuses on sprinkler irrigation practice, dealing with droplet aerial evaporation (i.e. the process of evaporation occurring to a water droplet when covering the path starting at the sprinkler nozzle outlet and ending ⇑ Corresponding author. Tel.: +39 0521905900. E-mail addresses: [email protected] (G. Lorenzini), [email protected] (O. Saro). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.02.062

before the soil is reached) and droplet general dynamics modelling. Such field was partially investigated in the last decades but still requires much work to discard any empiricisms in the full analytical description of the problem, allowing for a generally applicable solution. This issue is mainly due to the dependence of the thermal fluid dynamic phenomenon arising on a multi-parametric array affected by mutual interaction among its elements. The international literature panorama reflects some possible approaches to the assessment of droplet aerial dynamics and evaporation and to other related topics. A broad and still significant (also for the present investigation) literature review may be found in [19–21]: being these publications easily accessible also online, its content will not be repeated here to give space to further citations and deepenings, exception made for quoting the milestone researches by Edling [9], by Kincaid and Longley [15] and by Thompson et al. [34]. One of the first experimental–theoretical studies related to droplet evaporation that literature reports was that in [14]: aimed at linking experimental to analytical droplet evaporation, it considered water droplets of up to 2.6 mm diameter suspended from fine wires or glass filaments within a small chamber with controlled environmental conditions. Air was considered as still and the diffusion model employed made, in the own words of the author, the model not applicable to actual free-falling droplet problems, i.e. to the kind of problem tackled in this paper. Recently Bavi et al. [2] investigated the water spray losses experimentally in

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Nomenclature Notation A droplet external surface, m2 Acs droplet cross sectional area, mm2 Bi Biot number, – Cd friction factor (according to Fanning), – cpm specific heat of wet air (mixture), J kg 1 K1 Cs vapour concentration in air at the droplet surface (saturated air at droplet temperature), kmol m3 Cv vapour concentration in undisturbed air, kmol m3 cw specific heat of water, J kg1 K1 D droplet diameter, mm Di droplet initial diameter, mm Dv–a diffusion coefficient of water in air, cm2 s1 F generical function defining the state variables coefficients (Runge–Kutta method) Fi (i = 1, . . . , m) elements of F Fb buoyancy force, N friction force, N Ff Fg gravity force, N g acceleration of gravity, m s2 H nozzle height, m h convective heat transfer coefficient, W m2 K1 hm mass transfer convective coefficient, m s1 hr radiative heat transfer coefficient, W m2 K1 k friction parameter, kg m1 Kj,n (j = 1, . . . , 4) Runge–Kutta coefficients Isol solar irradiance, W m2 L droplet travel distance, m m droplet mass, kg Ma molar mass of air, [kg kmol1] mb net mass of the droplet (buoyancy effect in air), kg mev droplet mass evaporated in-flight, kg droplet mass evaporated in flight, % Mev Mm molar mass of wet air (mixture), kg kmol1 Mout mass of water evaporated per unit time, kg s1 mt mean evaporation velocity (=mev s1), kg s1 Mw molar mass of water (=18.02), kg kmol1 nm number of moles of wet air, – nv number of moles of vapour, – Nu Nusselt number, – Pr Prandtl number, – p atmospheric pressure (=101,325), Pa ps (Tw) vapour saturation pressure at water temperature, Pa vapour saturation pressure at air temperature, Pa ps (Ta) pv vapour pressure, Pa Qenv environmental radiation heat flux, W Qf frictional heat flux, W latent heat flux, W Ql Qs sensible heat flux, W Qsol solar radiation absorption, W r latent heat of evaporation, J kg1 Re Reynolds number, – R0 gas universal constant (=8314), J kmol1 K1

semi-portable hand move sprinkler irrigation systems for those climatic conditions typical of semi-arid regions, in presence of wind and of manifold operating conditions: their results confirmed the relevance of wind (i.e. of the air friction dynamics) on the aerial evaporation process, as in [20], arriving at some interesting empirical formulas (bond to the test conditions) linking evaporation to wind speed (i.e. scalar velocity) and to vapour pressure deficit. Also relevant to get insight of droplet size distribution in sprinkler irrigation practice is the recent study in [30], analysing the cumulative droplet frequency and volume in function of their

Sc Schmidt number, – Sh Sherwood number, – t time, s Ta air temperature, K Tw water temperature, K Twi initial droplet temperature, K U generic vector with the assigned inlet values Ui (i = 1, . . . , m) elements of U V droplet velocity, m s1 Vg droplet volume, m3 Vi droplet initial velocity, m s1 Vr relative droplet–air velocity, m s1 X generic vector of state variables X⁄ generic vector of state variables time derivatives Xi (i = 1, . . . , m) elements of X Xn+1 generic state variable solution x horizontal direction perpendicular to y y horizontal direction perpendicular to x ya molar fraction of dry air, – yv molar fraction of vapour, – z vertical direction positive upwards Greek notation a water absorptivity in relation to solar radiation (=0.94), – c wind velocity vector, m s1 d angle between initial droplet trajectory and horizontal direction, ° D operator of variation el droplet emissivity for long wavelength radiation (=0.95), – k air thermal conductivity, W m1 K1 l dynamic viscosity, m2 s1 m kinematic viscosity, cm2 s1 qa air density, kg m3 qm mixture (wet air) density, kg m3 qv vapour density, kg m3 qw water density (=1000), kg m3 r Stefan–Boltzman constant (=5.67  108), W m2 K4 s droplet time of flight, s u air relative humidity, % subscripts x x-direction component (horizontal) y y-direction component (horizontal and perpendicular to x) z z-direction component (vertical) 0 initial value superscript T transposed

diameter and of the distance from the nozzle droplet: the authors adopted a low-speed photographic technique tackling a sprinkler irrigation system provided with a 4.8 mm nozzle with impact arm and operating at a pressure of 200 kPa (which are among the typical in-field conditions). An interesting and recent experimental research was also made by Fujita et al. [11], who studied the air relative humidity effect on water droplet evaporation, confirming that the droplet temperature decreases in conditions of low air relative humidity and vice-versa, what the results presented in the present paper will also confirm. Interesting

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information and practical data related to evaporation in sprinkler irrigation practice are also available in [31] where climatic and operating conditions typical of Florida (USA) were investigated. Very important for the discussion of the results which this paper will present is the experimental investigation reported in [16]: it provides some experimental laser measurements of sprinkler droplet size distributions comparing them to other homogeneous literature date; the final results achieved prove that, for most of the typical operating conditions (nozzle dimension and working pressure), the water volume discharged is mainly composed of small diameter droplets: in most of the cases study reported, between 50% and 90% of the water volume discharged is to be attributed to droplets with a diameter of less than 2 mm; this percentage increases with working pressure and decreases with nozzle size. Such general trend gets significant elements of confirmation also in [7,17,10,33]. Shifting towards theoretical researches, recently Yan et al. [39] published a research paper comparing different modelling approaches related to sprinkler droplets dynamics, evaporation and distribution, before building up their own no-wind model. The main affecting parameters considered were: droplet diameter, relative humidity, nozzle height and air temperature. After introducing a detailed friction factor analysis, Park et al. [27,28]’s scheme was adopted. In particular: for Re 6 1000 ? Cd = (24/Re) (1 + 0.15 Re0.687) for Re > 1000 ? Cd = 0.438 {1.0 + 0.21[(Re/1000)  1]1.25}

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whole process; De Wrachien & Lorenzini [8] hypothesised a novel quantum approach to the dynamic description of moving water particles. The analytical model arrived at in the present paper and successively implemented by numerical means, makes many steps ahead with respect to a previous research by Lorenzini [20], 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 action; 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 fluid dynamic process becomes technically realistic. Moreover droplet dynamics and evaporation is modelled 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 of water in air, air relative humidity, droplet inlet inclination, solar and environmental radiation, wind vectorial velocity. Hence the process is here not just considered 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.

2. Analytical model 2.1. Hypotheses adopted

where Re [–] is Reynolds number and Cd [–] is friction factor. In the present paper anyway the classical Bird et al.’s [3] friction factor approach was adopted. Yan et al.’s [39] results confirm, among other results, the inverse proportionality between evaporation and droplet diameter and the role of air friction in sprinkler droplet evaporation. An important modelling contribution was also available in [18] even if aimed at considering a system interested by droplet dispersions and collisions, more typical of chemical systems or, in general, of clouds. Deserving to be mentioned also the research in [29], treating both experimentally and numerically the reciprocal effect of falling deformable droplets. Here the droplet deformation was not considered relying on the experimental evidence reported in [26]. In addition to other parameters already cited, also radiation plays an important role on droplet evaporation and on the joint dynamic process: in the present paper this has been considered, in relation to the heat flux associated, both as solar and as environmental radiation. The absorption coefficient of the solar radiation was assumed equal to 0.94 [–], which is typical of water for high frequency radiation [12]. A few deepenings about the effect of radiation on droplet evaporation are available in [35] and in [38]. Finally two well structured reviews may be quoted: in [36] the authors provide an overview of the existing panorama related to evaporation losses in sprinkler irrigation, describing and comparing many experimental and theoretical approaches. Basically trusting more theoretical than experimental methods – as the latter generally lead to results which may be strongly affected by measurement difficulties (which, they suggest, may be overcome using the eddy covariance technique) – they tend to prefer a ballistic–dynamic approach even if hoping for future improvements of the models available; in [13] it is also provided a complete survey on many topics related to spray systems: in-flight evaporation of droplets is also considered starting from Williamson and Threadgill’s [40] approach, successively tested through the code IDEFICS version 3.1. In conclusion two very recent contributions deserve a quotation: Molle et al. [25] performed an interesting experimental–numerical study on in-field water droplet evaporation during sprinkler irrigation, stressing out the importance of the droplet diameter and of the wind drift to assess the

As highlighted in the Introduction, the analytical model which will be defined in the following starts from and then completes the preliminary approach in [20]. This is why some of the hypotheses adopted there were held also here: (1) the physical system considered is a single droplet exiting from the nozzle of a sprinkler and generated exactly in correspondence to the nozzle outlet; (2) the forces applied to the system are weight, buoyancy and friction; (3) the droplet has a spherical shape all the way down [26]; (4) friction has the same direction of velocity for all the path but opposite sense; but, differently from Lorenzini [20]: (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 vectorial 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, without any simplifications in the thermal description of the analysis variables and none of the parametric effects is maximised – such as air friction in [20] – but always described in their true dynamics.

2.2. Analysis parameters and dynamic description The arbitrary parameters completing the description are (x is the horizontal direction covered by the droplet in absence of wind;

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y is the horizontal direction perpendicular to x; z is the vertical direction): (a) the nozzle is at ground level (height H = 0 m); (b) the droplet path considered is that associated to the trajectory range (i.e. when the vertical co-ordinate goes back to 0) (c) the modulus of the droplet initial velocity Vi (i.e. at the nozzle outlet) assumes 4 values: 5 m s1, 15 m s1, 25 m s1, 30 m s1; (d) the inlet inclination (with respect to the horizontal direction) of the droplet path d assumes 5 values: 10°, 25°, 40°, 55°, 70°; (e) the droplet initial diameter Di assumes 9 values: 0.1 mm, 0.4 mm, 0.7 mm, 1 mm, 2 mm, 3 mm, 4 mm, 5 mm, 6 mm; (f) the wind velocity c assumes 21 values (8 of which, the vertical ones, are not as realistic as the others): 0 m s1, 1 m s1, 2 m s1, 3 m s1, 4 m s1 in three directions and two senses (symmetrical cases already excluded; the wind itself may create a non-stationary turbulent boundary layer near the land surface, and air could be filled also with dust particles of different sizes: this was here not considered); (g) air temperature Ta assumes 5 values: 300 K, 305 K, 310 K, 315 K, 320 K; (h) initial droplet temperature Twi assumes 5 values: 288 K, 296 K, 304 K, 312 K, 320 K; (i) relative humidity u assumes 5 values: 0%, 10%, 20%, 50%, 70%; (j) diffusion coefficient of water in air Dva assumes 5 values: 0.200 cm2 s1, 0.225 cm2 s1, 0.250 cm2 s1, 0.275 cm2 s1, 0.300 cm2 s1; (k) solar irradiance Isol assumes 5 values: 0 W m2, 250 W m2, 500 W m2, 750 W m2, 1000 W m2. The forces acting upon the droplet in flight are, in the present approach (vectors are in bold font, on the contrary of their moduli): (I) Fg = mg (gravity) directed vertically and oriented downwards as z where z is the vertical axis positive upwards (m is droplet mass, g is acceleration of gravity); (II) Fb = qap(D3/6) g (buoyancy) directed and oriented as +z (qa is air density, D is droplet diameter); (III) Ff = kVrVr (friction) having same direction but opposite sense with respect to the relative droplet–air velocity Vr; k = (1/2) Cdqa Acs is the friction parameter, Acs is the cross sectional area of the droplet, Cd = f(Re) is the dimensionless friction factor according to Fanning [3], qa is again air density. In particular: Vr = V  c and:

Vr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV x  cx Þ2 þ ðV y  cy Þ2 þ ðV z  cz Þ2 ;

where V is droplet velocity; c is wind velocity, (cx, cy, cz) are its components and (Vx, Vy, Vz) are the droplet velocity components. 2.3. Equations of motion According to the international thematic literature [6,22,5] the study of the order of magnitude of the forces acting on particles based on the equation of motion derived by Maxey and Riley [23] reveals that the virtual mass and the pressure gradient are O(qa/qw) and the Basset force is O[(qa/qw)1/2]. In this work (qa/ qw) is O (103). Thus for the specific system examined here, neglecting the virtual mass and the Basset force have a limited effect and the equation of motion reduces to a balance of Stokes drag,

buoyancy forces and particle inertia. The equations describing how a water droplet moves in its aerial path from the sprinkler nozzle to the ground are set, accordingly to the forces list in the previous section, as:

  dx ¼0  c x 2 dt dt   2 d y dy  cy ¼ 0 m 2 þ kV r dt dt   2 d z dz  cz ¼ mb g m 2 þ kV r dt dt 2

m

d x

þ kV r

ð1Þ ð2Þ ð3Þ

being c the wind velocity vector and (cx, cy, cz) its scalar components, m the droplet mass and mb the net droplet mass as diminished by buoyancy effect; which writing the droplet velocity components as:

Vx ¼

dx ; dt

Vy ¼

dy ; dt

Vz ¼

dz dt

allows one to write the following differential equations system to be solved:

dV x þ kV r ðV x  cx Þ ¼ 0 dt dV y m þ kV r ðV y  cy Þ ¼ 0 dt dV z m þ kV r ðV z  cz Þ ¼ mb g dt

m

ð4Þ ð5Þ ð6Þ

2.4. Mass transfer Mass transfer within the process analysed is described by means of vapour concentration in air as:

dm ¼ hm AMw ðC v  C s Þ dt

ð7Þ

where A is droplet external surface; hm is mass transfer convective coefficient; Cs is vapour concentration in air at the droplet surface in a condition of saturated air at droplet temperature, Mw is the molar mass of water (equal to 18.02 kg kmol1) and Cv is vapour concentration in undisturbed air. Considering vapour as an ideal gas one may write: C s ¼ pRs0ðTTwwÞ and C v ¼ Rp0 vT a , being pv = u  ps(Ta) the vapour pressure within the system, ps is the saturation pressure of water at air temperature, u the relative humidity and R0 the gas universal constant (8314 J kmol1 K1). The mass transfer convective coefficient hm [m s1] may be obtained by Sherwood number Sh [3]:

Sh ¼

1 1 hm D ¼ 2 þ 0; 6Re2 Sc3 Dv a

ð8Þ

where Sc is Schmidt number Sc ¼ Dvma . In particular: D is droplet diameter, Dva is diffusion coefficient of water in air, Re ¼ Vmr D is Reynolds number (with Vr relative air–droplet velocity, defined in the previous section of this paper). Droplet diameter and mass are strictly related: m = qwVg being qw (=1000 kg m3) water density 3 and V g ¼ p6D droplet volume. The saturation pressure of water is evaluated (both for vapour in air and for the droplet) as [1]:

lnðps Þ ¼

c1 þ c2 þ c3 T þ c4 T 2 þ c5 T 3 þ c6 lnðTÞ; T

ð9Þ

where T [K] is temperature at which ps [Pa] is evaluated (Tw or Ta) and, for T > 273.15 K, c1 = 5800.2206 [K] c2 = 1.3914993 [–] c3 = 0.048640239 [K1] c4 = 4.1764768  105 [K2] c5 = 1.4452093  108 [K3] c6 = 6.5459673 [–].

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2.5. Heat transfer The droplet in flight is interested by both a sensible and a latent heat transfer process. The former, Qs, is to be written as follows:

Q s ¼ hAðT a  T w Þ where h is the convective heat transfer coefficient which may be computed by: 1 1 Nu ¼ hD ¼ 2 þ 0; 6Re2 Pr 3 with k thermal conductivity of air (a k constant equal to 0.026 W m1 K1); Re is Reynolds number; Pr is Prandtl number, computed holding the value of l (dynamic viscosity) coincident with that of dry air at a temperature of 310 K, i.e. equal to 1.67  105 m2 s1; while the wet air (mixture) specific ðc q þc q Þ heat cpm is variable and equal to cpm ¼ pa aq pv v , where qv is vam pour density defined as the ratio between vapour mass and wet air volume and where, hypothesising both dry air and vapour as m ideal gases, the mixture (wet air) density qm becomes: qm ¼ pM ; R0 T a with the molar mass of wet air (mixture) Mm defined as weighted mean of the molar masses of both components: Mm = Mwyv + Maya, in which Ma is the molar mass of air, yv ¼ ppv ¼ nnmv is the molar fracv ¼ nm nv is the molar fraction of dry air and tion of vapour, ya ¼ pp p nm p = 101325 Pa is atmospheric pressure (nv = number of moles of vapour; nm = number of moles of wet air). The latent heat transfer, Ql, may be written as:

Q l ¼ rM out ¼ r

dm dt

where r is latent heat of evaporation and Mout is the mass of water evaporated per unit time. In addition [20], there is also a frictional heat flux contribution Qf, due to the friction force acting upon a water droplet in flight because of the presence of air:

Q f ¼ Ff V r where Ff is the friction force module and Vr is the module of the relative air–droplet velocity. Finally radiation is considered as solar and environmental components. The solar radiation absorption Qsol was accounted for as:

Q sol ¼ Isol aAcs where Isol is solar irradiance, a = 0.94 [–] is water absorptivity in relation to solar radiation [12], Acs is the cross sectional area of the droplet. The environmental radiation heat flux Qenv is written as:

Q env ¼ hr AðT a  T w Þ T a þT w 3

where hr ¼ 4rel 2 is the radiative heat transfer coefficient, r = 5.67  108 W m2 K4 is the Stefan–Boltzman constant, el = 0.95 [–] is droplet emissivity for long wavelength radiation [12]. After preliminary tests, the effect of Qenv on droplet evaporation proved to be negligible and was not considered further. These fluxes added together are equal to the droplet internal energy variation that, considering the droplet as an isothermal system (Bi < 0.1), can be written as:

cw m

dT w ¼ Q s þ Q l þ Q f þ Q sol þ Q env dt

where cw is the specific heat of water, Tw its temperature and m the droplet mass. The Biot number is always l.t. 0.1 for the whole range of cases tested, thus the assumption of isothermal droplet is here well posed and also accounts for the contribution of the internal droplet mixing happening during the in-flight path [35]. Instead, in case the droplet internal temperature profile is investigated, other approaches may be followed [24]. Consequently, the analytical model defined allows one to describe the actual process occurring to a water droplet travelling in air from a sprinkler nozzle to the ground, i.e. taking into account the whole set of parameters that

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play a role in the actual process itself: droplet diameter, flow state, air temperature, water temperature, initial and step-by step droplet velocity, droplet–air friction, water–air diffusion coefficient, vapour concentration in air, solar radiation. In particular, the flow state is to be monitored carefully as the friction parameter k, defining how friction affects the droplet path, is bond to the flow state due to the value of Cd [3]:  (Stokes’ law flow) for Reynolds number Re < 0.1 ! C d ¼ 24 ; Re  (intermediate law flow) for 2 < Re < 500 ? Cd = 18.5 Re(3/5);  (Newton’s law flow) for 500 < Re < 200,000 ? Cd = 0.44. All these options were obviously implemented in the numerical code defined in the next section of this paper to assess the process evolution during the droplet aerial path. The tests performed for this paper were 64, 9 of which duplicated as a consequence of the overlap among the analysis parameters. The parametric study was made based on the classic approach of testing the variables one at a time after having fixed to a previously decided value all the other parameters. The whole set of cases study are reported in Table 1. The values of Nu and Sh computed throughout the simulations comply with the conditions for applying the Chilton– Colburn similarity (see [3]), used to derive the correlations for Sh and Nu. Throughout the computations performed it was employed a constant value of the film Pr equal to 0.7307 with an error on heat transfer O[(Pr/Prf)1/3], always l.t. 0.4%; and a constant value of the film Sc equal to 0.6016 with an error on mass transfer O[(Sc/Scf)1/3], always l.t. 1.3%. 3. Numerical approach Many methods are nowadays available to perform numerical simulations implying different levels of complication: more elaborated methods provide better results in terms of stability and of accuracy but generally with the drawback of an elevate simulation time. The problem here investigated is referable to a Cauchy one, being known the initial conditions and the differential equations ruling the process. In general, a Cauchy problem may be outlined as:



X ðtÞ ¼ Fðt; XðtÞ; UðtÞÞ Xðt 0 Þ ¼ X 0

with

8 XðtÞ ¼ ½X 1 ðtÞ;X 2 ðtÞ;X 3 ðtÞ;...X m ðtÞT ; > > > > > < Fðt;XðtÞ;UðtÞÞ ¼ ½F 1 ðt;X;UÞ;F 2 ðt;X;UÞ;F 3 ðt;X;UÞ;...;F m ðt;X;UÞ;T UðtÞ ¼ ½U 1 ðtÞ;U 2 ðtÞ;U m ðtÞ;...;U m ðtÞ;T > > > h i > > : X ðtÞ ¼ dX1 ðtÞ ; dX2 ðtÞ ; dX 3 ðtÞ ;...; dX 3 ðtÞ ; T dt

dt

dt

dt

where X(t) is the state variables vector and U(t) is the vector with the assigned inlet values. In the present case the state variables are: x, y, z (position) and their first derivatives; m (mass); Tw (droplet temperature). Applying a Runge–Kutta fourth order method then, for each state variable, the solution becomes:

Xnþ1 ¼ Xn þ

Dt ðK 1;n þ 2K 2;n þ 3K 3;n þ 4K 4;n Þ 6

where the coefficients Kj,n (j = 1, . . . ,4) may be evaluated at every time step as:

8 K 1;n > > > < K 2;n > K 3;n > > : K 4;n

¼ FðnDt; X n Þ   ¼ F nDt þ D2t ; X n þ D2t K 1   ¼ F nDt þ D2t ; X n þ D2t K 2 ¼ FðnDt þ Dt; X n þ DtK 3 Þ

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Table 1 Cases study considered (the corresponding case numbers of those duplicated are highlighted in dark grey; the parameters variations are highlighted in light grey). Cases study Case number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Analysis parameters Di [mm]

Vi [m s1]

cx [m s1]

cy [m s1]

cz [m s1]

di [°]

Ta [K]

Dva [cm2 s1]

u [%]

Twi [K]

Isol [W m2]

0.1 0.4 0.7 1 2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

30 30 30 30 30 30 30 30 30 5 15 25 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 2 0 0 0 3 3 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 2 0 0 0 3 3 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 10 25 40 55 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70

320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 300 305 310 315 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320 320

0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.200 0.225 0.250 0.275 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 20 50 70 0 0 0 0 0 0 0 0 0 0

304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 304 288 296 304 312 320 304 304 304 304 304

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 0 250 500 750 1000

With this algorithm the function F is evaluated 4 times per time interval, once at the beginning, twice in the intermediate point and once again at the end of the interval; the four values thus obtained are linearly combined for calculating the new value Xn+1. Several assessments of F are required for applying the Runge–Kutta

fourth order method and hence it appears that a higher degree of accuracy means more complicated calculations: this is, in general, true unless the algorithm accuracy is the priority. To implement the above described model it was employed Microsoft Excel. It is beyond doubt that Excel is a fast and easy-to-use electronic sheet

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application: its appealing interface and multifarious functions, with the integration of Visual Basic for Applications (VBA) programming language, make of it a powerful and versatile tool. In addition it is extremely popular and easily accessible. Consequently the problem solutions were arrived at by means of the VBA functions. Such approach proved to be effective, despite being quite time consuming [4].

4. Results and comments The tests performed allowed for the collection of many data related to the complicate dynamic and thermal fluid dynamic phenomenon examined. The first and more evident result to be assessed is that concerning the in-percentage droplet mass evaporated during the aerial path Mev as a function of each variable. The trends obtained are displayed in the figures from 1 to 9. Even at first sight the effect of the solar radiation absorption Qsol on the droplet mass evaporation proves absolutely negligible despite Qsol itself changes of 3 orders of magnitude during the tests within the typical range of variation (Fig. 1): this is to be attributed to the small entity of such heat fluxes with respect to Qs and Ql, that is to the sensible and latent heat fluxes, respectively. Similarly (Fig. 2), also the wind velocity c appears not to affect droplet evaporation significantly (the figure reports x- and z- components) as its effect is nearly entirely dynamic (i.e. affecting the droplet trajectories) but its contribution to Qf, which itself is a small part of the total heat flux, is limited also because the value of Vi set for the cases 14  34 (see Table 1) is always much bigger than c, as it has to be to keep the study tight to an actual condition. On the contrary, all the other parameters investigated reveal a relevant influence on Mev. The droplet initial diameter Di, in particular, has a dramatic effect on evaporation, especially for small values (Fig. 3; cases 1  9; fixed parameters: Vi = 30 m s1; d = 70°; u = 0%; Dva = 0.3 cm2 s1; Twi = 304 K; Ta = 320 K): nearly 45% of the initial mass evaporates for Di = 0.1 mm and still more than 6% for a diameter which is of an order of magnitude bigger. This result is particularly important if one thinks that a sprinkler water jet, mainly according to [16] but also to other scientists [17,33,7,30,10], is in most cases composed from 50% to 90% of its volume by droplets with a diameter of less than 2 mm: this means that the relative weight of small droplets evaporation is to be carefully accounted for as playing a very significant role in the process. Going ahead, while on the one hand the influence of the air temperature on evaporation (Fig. 4; cases 40  44) could have been somehow qualitatively predictable (resulting useful as an indirect form of validation of the model created), on the other an increase of less than

Fig. 1. Mev [%] vs. Qsol [W].

329

1.2% in droplet evaporation (fixed parameters: Di = 1 mm; Vi = 30 m s1; d = 70°; u = 0%; Dva = 0.3 cm2 s1; Twi = 304 K) when passing from an air temperature value of 300 K to 320 K seems somehow surprisingly low and it may be attributed to the very quick dynamics of the process which affects the thermal transient state within the droplet. Similarly (Fig. 5; cases 45  49; fixed parameters: Di = 1 mm; Vi = 30 m s1; d = 70°; u = 0%; Ta = 320 K; Twi = 304 K), apart from being qualitatively obvious, it results astonishing that an increment of so much as 50% in the diffusion coefficient Dva determines an evaporation increase of less than 0.9%. In this sense the initial water temperature Twi proves to be much more effective in influencing the phenomenon (Fig. 6; cases 55  59; fixed parameters: Di = 1 mm; Vi = 30 m s1; d = 70°; u = 0%; Dva = 0.3 cm2 s1; Ta = 320 K): passing from 288 K (typical of water from the water supply network) to 320 K, the evaporation nearly doubles shifting from a bit more than 4% to a bit less than 8%. This datum is extremely interesting as it provides a practical suggestion (which should be taken into consideration) for water saving in irrigation during the hot season: it is more ecological – in the sense of the present investigation – to irrigate with tap water rather than with water stored in reservoirs or small artificial basins. The affection of the droplet initial velocity Vi [m s1] on its evaporation was also tested (Fig. 7; cases 10  13; fixed parameters: Di = 1 mm; Twi = 304 K; d = 70°; u = 0%; Dva = 0.3 cm2 s1; Ta = 320 K): higher values produce higher evaporation (two and a half times more passing from 5 to 30 m s1) due to a more

Fig. 2. Mev [%] vs. c [m s1].

Fig. 3. Mev [%] vs. Di [mm].

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Fig. 4. Mev [%] vs. Ta [K].

Fig. 5. Mev [%] vs. Dva [m2 s1].

significant effect of air friction and a longer s [s] so discouraging, for a same initial amount of water employed, irrigation policies based on high pressure-high range sprinkler plants. The effect of d on droplet evaporation is to be studied as associated to the corresponding time of flight trends, as Fig. 8 displays (cases 35  39; fixed parameters: Di = 1 mm; Vi = 30 m s1; Twi = 304 K; u = 0%; Dva = 0.3 cm2 s1; Ta = 320 K). In fact if an increase of d obviously (within a certain range) corresponds to an increase of s [s] and consequently of Mev [%], the trend of Mev [%] versus d [°] proves to be less steep than that of s [s] versus d [°], meaning that less inclined jets tend to be interested by a proportionally bigger evaporation. Completing the first part of this discussion, droplet evaporation Mev obviously decreases with the relative humidity u but, what it is interesting to be observed (Fig. 9; cases 50  54; fixed parameters: Di = 1 mm; Vi = 30 m s1; Twi = 304 K; Ta = 320 K; Ta = 320 K), it drops to 0 for a value of u between 60% and 70% (and not for u = 100% as one could apparently expect) and the water droplet raises its mass from that condition on, with negative values of Mev [%]. This is because for elevate values of u the saturation pressure ps (Tw) results to be lower than pv so causing water condensation on the droplet surface. Successively the latent heat of condensation received by the droplet, together with the heat conduction due to the hotter surrounding air, determines an increase of temperature Tw, i.e. of ps (Tw), which at a certain point becomes greater than pv so triggering evaporation. The full picture of the relation between vapour pressure pv and droplet saturation

Fig. 6. Mev [%] vs. Twi [K].

Fig. 7. Mev [%] vs. Vi [m s1].

Fig. 8. Mev [%] vs. d [°].

pressure ps (Tw) is showcased in Fig. 10, displaying the trend of ps(Tw) with time adopting the relative humidity u as parameter, i.e. highlighting for the cases 50  54 the trend of ps (Tw) as compared to that (constant case by case) of pv. For the lower relative humidity cases (0%, 10%, 20%) the ps (Tw) curve is always higher than the pv one and this means that the pressure gradient is in full favour of a droplet evaporation, while in the higher relative

G. Lorenzini, O. Saro / International Journal of Heat and Mass Transfer 62 (2013) 323–335

Fig. 9. Mev [%] vs. u [%].

Fig. 10. ps (Tw) [Pa] vs. t [s], u [%] parameter.

Fig. 11. mt [kg s1] vs. Di [mm].

humidity cases (50%, 70%) there is a split trend, being the ps (Tw) curve higher than the pv one just in a part of the whole trend representing the phenomenological evolution. Consequently in a part of the process, namely at the beginning, the droplet will receive mass from water vapour in the air condensing on its surface, this process going into reverse afterwards, when the ps (Tw) values

331

Fig. 12. mt [kg s1] vs. Vi [m s1].

Fig. 13. mt [kg s1] vs. c [m s1].

exceed the pv one. The bigger u the more vapour condensation will overtake droplet evaporation but even significantly lower than 70% values of u causes a globally negative value of Mev. In addition to the in-percentage water mass evaporated Mev, it is also important to consider the effect on mean evaporation velocity mt (namely: mt = mev s1) of the analysis parameters. The corresponding trends are on display in figures from 11 to 19. If the droplet initial diameter Di is the analysis parameter, the corresponding mean evaporation velocity is reported in Fig. 11 showing a trend which is qualitatively opposite to that of Fig. 1: the reason is that the mass of water evaporated mev augments of 4 orders of magnitude while the time of flight s just increases fourfold. Needing some words of comment is also the trend of mt with the droplet initial velocity (Fig. 12): even if quantitatively the global shift is very limited, qualitatively its trend is decreasing because the time of flight augments with Vi more than mass evaporation. A similar trend (see Fig. 13) is observed when evaluating the effect of wind velocity components on mean evaporation velocity: the highest mt values are recorded with the lowest (highest with negative sign) values of cx as the relative velocity of the system is maximum; the transversal z component cz does not show a significant trend. Initial jet inclination effect (Fig. 14) on mt is also explainable as mev almost doubles but s almost quadruplicates. The trends in Figs. 15–18 are linearly increasing. For what relates to Fig. 19 (the effect of u on mt) a more humid environment tends to slow evaporation (successively stopping it and the even inverting the process, as already

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Fig. 14. mt [kg s1] vs. d [°].

Fig. 17. mt [kg s1] vs. Twi [K].

Fig. 15. mt [kg s1] vs. Ta [K].

Fig. 18. mt [kg s1] vs. Qsol [W].

Fig. 16. mt [kg s1] vs. Dva [m2 s1].

Fig. 19. mt [kg s1] vs. u [%].

commented) because it reduces the vapour pressure gradient between droplet and air. It is also interesting to carry out a thermal assessment of the tests performed, checking what happens to the heat fluxes involved in the process and exchanged between the droplet and the local environment. A few significant cases were selected as qualitatively representative of many others (Fig. 20,

case 64) or as allowing for significant comments (Fig. 21, cases 50 – coincident with case 4 – and case 53). The first consideration is that the heat fluxes due to radiative and frictional contributions are rather small with respect to those due to convective and latent contributions (see Fig. 20) and will not be deepened here. Fig. 20 also displays the close relation between the dynamic and the

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to 0% and 50%, respectively. As one may appreciate, while for convective heat flux the global trend is very similar to that in Fig. 20, the situation is different for the latent heat fluxes: Q1 is always negative for the former case – and this proves that the droplet is evaporating throughout the test – while it is initially positive and just afterwards becomes negative for the latter, meaning that while initially the droplet receives energy by the water vapour in air condensing on its surface, successively the process comes to its opposite: this confirms once more the complexity of the phenomenon examined which is strictly dependent on the whole range of parameters defining the system investigated.

5. Validation

2

Fig. 20. Case 64: thermal fluxes per unit area [W m

] and Re [–] vs. t [s].

Fig. 21. Case 50 (u = 0%) and 53 (u = 50%): thermal fluxes per unit area [W m2] vs. t [s].

thermal fluid dynamic aspects of the process investigated, highlighted by the fact that maximum, minimum and increasing/ decreasing trends of the heat flux exchanged are strictly related to the trend of Re, which is absolutely evident for what concerns convective heat flux but which becomes also logical for the latent heat flux once one remembers that the mass transfer coefficient hm (see Eq. (8)) is dependent on Re. Fig. 21, as specified above, displays the convective and latent heat fluxes trends versus time as referred to cases 50 and 53 in Table 1, i.e. when relative humidity u is equal

The whole set of results presented in the previous section proved how effective and sound is the model presented in this paper and how much it may help the full understanding of the process in exam. It is, anyway, extremely useful to assess if the computed data match with those available in well established literature studies. Aiming at this we considered again the researches by Edling [9] and by Thompson et al. [34], focusing both on droplet kinematic (travel distance L and time of flight s) and on droplet evaporation (droplet mass evaporated in flight Mev [%]) results. Tables from 2 to 6 report those comparisons. The analysis parameters tested were: – Edling [9]:  kinematic analysis: H = 1.22 m; 2.44 m; 3.66 m; evaporation analysis: 3.66 m;  kinematic analysis: Ta = 302.55 K; evaporation analysis: Ta = 294.26 K;  kinematic analysis: d = 10°; 0°; 10°; evaporation analysis: d = 0°;  kinematic analysis: Vi = 11.37 m s1; evaporation analysis: Vi = 18.23 m s1;  kinematic analysis: Di = 0.5 mm; 1.5 mm; 2.5 mm; evaporation analysis: Di = 1 mm; 1.125 mm; 1.25 mm; 1.375 mm; 1.5 mm;  u = 20%  c = 0; – Thompson et al. [34]:  H = 4.5 m;  Ta = 311.15 K  d = 25°;  Vi = 30.91 m s1  Di = 0.3 mm; 0.9 mm; 1.8 mm; 3 mm; 5.1 mm;  u = 20%  c = 0.

Table 2 Travel distance: Edling’s [9] vs. present work results. H [m]

d [°]

L [m] Di [mm] 0.5

1.5

2.5

Edling

This work

Edling

This work

Edling

This work

1.22

10 0 10

1.53 1.52 1.46

1.35 1.35 1.31

4.04 3.55 2.91

4.39 3.78 3.03

5.08 4.19 3.22

5.34 4.34 3.33

2.44

10 0 10

1.55 1.55 1.50

1.36 1.36 1.32

4.62 4.31 3.86

5.07 4.67 4.11

6.00 5.37 4.57

6.34 5.61 4.74

3.66

10 0 10

1.55 1.55 1.50

1.36 1.36 1.32

4.95 4.73 4.36

5.48 5.19 4.73

6.60 6.10 5.41

7.00 6.40 5.63

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Table 3 Travel distance: Thompson et al.’s [20] vs. present work results. Di [mm]

0.3 0.9 1.8 3.0 5.1

6. Conclusions

L [m] THOMPSON et al.

This work

1.30 5.22 10.00 13.48 17.83

0.95 5.24 10.89 15.71 22.25

Table 4 Time of flight: Thompson et al.’s [34] vs. present work results. Di [mm]

0.3 0.9 1.8 3.0 5.1

s [s] Thompson et al.

This work

2.63 1.54 1.63 1.75 1.84

0.56 1.01 1.24 1.46 1.68

Table 5 Droplet evaporation: Edling’s [9] vs. present work results. Di [mm]

Mev [%] Edling

This work

1 1.125 1.25 1.375 1.5

1.19 1.08 1.01 0.95 0.81

2.08 1.80 1.58 1.40 1.26

Table 6 Droplet evaporation: Thompson et al.’s [20] vs. present work results. Di [mm]

1 1.125 1.25 1.375 1.5

Mev [%] Thompson et al.

This work

2.39 2.11 1.85 1.75 1.41

2.96 2.48 2.13 1.86 1.65

For what pertains the kinematic analysis, the travel distance results show a general good agreement with the reference data, highlighting a slight underestimation (a few percent) of L for small Di values and a slight overestimation for large Di values, but all the trends are respected: this holds true both for Edling’s [20] and for Thompson et al.’s [34] data (see Tables 2 and 3). This clearly proves the reliability of the model presented and the small discrepancies, not in the trends but just in the figures, can be easily ascribed to the different modelling approach adopted as, for instance, neither Edling [9] nor Thompson et al. [34] considered air friction among the affecting parameters. Still on kinematics, also the time of flight results (see Table 4) show a good agreement and the same trend as in Thompson et al. [34], except for the case with Di = 0.3 mm, that anyway seems not sound among the reference data, thus not affecting the general evaluation. The evaporation comparisons (Tables 5 and 6) also show a very good agreement with a matching general trend and a slightly increased evaporation in our computed data, again due to the more complete set of affecting variables adopted. And this latter comparison is particularly significant as the Di [mm] values tested are among those statistically more probable in an agricultural sprinkler jet ([30]).

The struggle for fresh water resources hoarding will probably characterise the next decades. Thus a sustainable water use policy is by now necessary and useful. This will have to do especially with agriculture as it is in such sector that water depletion is particularly evident and must be reduced and rationalised. In water irrigation practice, consequently, the most interesting savings, for what pertains water resources, could be done. The present paper has faced the problem of a complete theoretical description of a sprinkler water droplet leaving the nozzle and travelling in air undergoing a partial evaporation due to the many geometric, dynamic and thermal parameters affecting both the system and the process. The mutual interaction between the parameters involved makes it extremely hard to attain such results, as the very few similar attempts available in literature prove, albeit a relevant number of related papers have been written so far. In particular the study variables (within a sufficiently broad range) were: initial droplet diameter, initial droplet velocity, initial water temperature, air temperature, wind velocity, initial droplet trajectory inclination, solar radiation absorption, diffusion coefficient of water in air, air relative humidity; while the model considered all the thermal fluid dynamics transient implications (heating, chilling, evaporation, condensation) and the simply dynamic ones (friction, buoyancy). The application of a Runge–Kutta fourth order method to the model, allowed for the writing of a numerical code of general applicability to the different practical situations possible and to the full testing of the effect of each variable on the whole process. In detail, 64 numerical tests were made for the present study, highlighting that: initial droplet diameter has the broadest influence on evaporation: more than 44% of the initial mass evaporates for Di = 0.1 mm and still more than 6% for a 1 mm diameter droplet, being this extremely important as a sprinkler water jet is mainly composed by droplets with a diameter of less than 2 mm; solar radiation absorption proves absolutely negligible; so does wind velocity, as its effect is nearly entirely dynamic; air temperature and diffusion coefficient also prove to be effective parameters, but they influence droplet evaporation less than preliminary predictable (about 1% in the range examined); initial water temperature, surprisingly, proves to be much more effective, so providing the practical suggestion not to store irrigation water, during the hot season, in basins or reservoirs; a similar consideration is good also for the initial droplet velocity effect, discouraging high pressure-high range irrigation plants; with reference to the initial jet inclination and suitably weighting it with the corresponding time of flight, the model also shows that less inclined jets tend to be interested by a proportionally bigger evaporation; while a raise in air relative humidity stops droplet evaporation between 60 and 70%, in the test conditions, and not 100% as one may expect, showing also transient states of condensation depending on the vapour pressure gradient. The latter relation was broadly examined in a previous section of this paper. Mean evaporation velocity was also assessed in relation to the analysis variables, together with the interactions between kinematic and heat flow parameters. A kinematic (travel distance and time of flight) and thermal fluid dynamic (in-percentage evaporation) validation of the model was also provided, using the classical data by Edling [9] and by Thompson [34], so proving the reliability of the model defined. Acknowledgements The authors are grateful to the Italian Ministry for Agricultural Alimentary and Forestry Policies which funded this research.

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