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Dynamic modelling and simulation of a heated brine spray system Raquel Durana Moita a , Henrique A. Matos a,∗ , Cristina Fernandes a , Clemente Pedro Nunes a , Mário Jorge Pinho b a b

Instituto Superior Técnico – DEQB, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Grupo CUF, Quinta da Indústria, 3864-755 Estarreja, Portugal

a r t i c l e

i n f o

Article history: Received 1 August 2008 Received in revised form 11 December 2008 Accepted 29 January 2009 Available online 7 February 2009 Keywords: Dynamic modelling Simulation Spray system Brine Ballistics theory gPROMS Evaporative process

a b s t r a c t The aim of this work is to build a model of a heated brine spray system in order to predict its behaviour through dynamic simulation and thus optimize its performance. Concentrated brine solutions with initial temperatures from 65 to 85 ◦ C are sprayed into the surrounding ambient air, in windy conditions. This spray system will be used in the NaCl salt recrystallization ponds that are integrated with a cogeneration unit, in order to increase the global process efﬁciency. This industrial platform is located in Pombal, Portugal, as already referred in a previous work [Moita, R. D., Matos, H. A., Fernandes, C., Nunes, C. P., & Prior, J. M. (2005). Dynamic modelling and simulation of a cogeneration system integrated with a salt recrystallization process. Computers and Chemical Engineering, 29, 1491–1505]. A global three-dimensional mathematical dynamic model was built, which includes two models. The single drop model, which is based on the ballistics theory and includes material and energy balances, allows calculating each drop trajectory and velocity as it exits the nozzle, as well as its temperature, salt concentration and volume. The spray system model accounts for the full-cone spray-nozzle by considering a set of random deﬁned drops. The spray system model was implemented and simulated in gPROMS 2.3.7. A sensitivity analysis of some model parameters and choices was performed. Model predictions, obtained through dynamic simulation, were compared with retrieved literature data, referring to water drops, in terms of drop trajectories and of evaporation rates. The model validation was also carried out using the experimental data obtained in the hereinabove mentioned NaCl recrystallization industrial ponds, referring to the respective heated brine spray system. It must be stressed that the simulated results are in reasonable agreement with the literature values and also with the experimentally measured ones. Although the model developed in this work allows predicting the behaviour of the mentioned heated brine scheme, it can be easily adapted to account for other types of systems and sprays. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Spray systems consist in a set of nozzles used in the formation of the spray (liquid dispersed as a stream of droplets), allowing to increase the liquid surface area and therefore to enhance evaporation, as well as to distribute the liquid over an enlarged area. Spray systems are very important and widely used in several engineering processes and industrial applications, such as, in sprinkler irrigation and pesticides spraying, spray drying, ﬁre extinguishment, evaporative cooling, dust control, and so on. Sprays are commonly used in agricultural irrigation, in which water with temperature values near ambient air temperature is sprayed to the surrounding air in windy conditions, through the use of sprinklers. There are available in the literature several mod-

∗ Corresponding author. Tel.: +351 21 841 7639; fax: +351 21 841 7638. E-mail address: [email protected] (H.A. Matos). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.01.016

els that were developed to study and optimize irrigation systems. A good summary of these models, in which evaporation and drift losses are accounted for, is presented by Wrachien and Lorenzini (2006). Existing studies are divided into two approaches: the statistical approach, in which measured evaporation losses relates to environmental and operational parameters, and the physical mathematical approach, which resorts to models that link equations ruling water droplet evaporation with particle dynamics theory. Studies related to improvement on experimental techniques, designed to reduce measurements errors, are also analyzed. Following the physical mathematical approach, several models are available for irrigation simulation purposes. Edling (1985) model, which is based on the impulse momentum principle, allows estimating kinetic energy, evaporation and wind drift of droplet from low-pressure irrigation sprinklers, to study the inﬂuence of design and atmospheric parameters on droplet behaviour. Thompson, Gilley, and Norman (1993) proposed a model to predict

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Nomenclature drop surface area (m2 ) oriﬁce area (m2 ) drag coefﬁcient mean heat capacity (J/(kg K)) air heat capacity (J/(kg K)) drop diameter (m) Rosin-Rammler parameter that corresponds to mean diameter (m) d0 oriﬁce diameter (m) Dab mass diffusivity of water vapour though air (m2 /s) Econv energy loss by convection (J/s) Eff global evaporative spray efﬁciency enthalpy of the NaCl salt precipitated in the drop ENaCl (J/s) Erad energy loss by radiation (J/s) solar energy absorbed through the brine solution in Esolar the drop (J/s) enthalpy of a vapour water stream (J/s) Evap fabs hemispherical absorptivity g gravitational acceleration (m/s2 ) G solar radiation that reaches the earth’s surface (J/(s m2 )) hc average convective heat transfer coefﬁcient (W/(m2 K)) hm average convective mass transfer coefﬁcient (m/s) nozzle height with respect to ground level (m) h0 Hbrine total enthalpy of the brine solution inside the drop (J) 25 ◦ C heat of dissolution of the salt in the water (J/kg NaCl) Hdiss air conductivity (W/(m K)) kair K Von Karman constant brine characteristic spray constant Kbrine Kwater water characteristic spray constant m Rosin-Rammler parameter that relates to the spread particle size total mass of the brine in the drop (kg) mbrine mdrop total drop mass (kg) mNaClpp,total total salt mass precipitated in the drop (kg) Mbrine,sp total spray brine mass by volume unit of entering brine (kg/m3 ) Mevap water evaporation rate in the drop (kg/s) Mevap,sp total water evaporated in the spray by volume unit of entering brine (kg/m3 ) MNaClpp salt precipitation rate in the drop (kg/s) MNaClpp,sp mass of precipitated salt in spray by volume unit of entering brine (kg/m3 ) n number of moles (moles) Nu Nusselt number P liquid stream differential pressure (bar) Pr Prandlt number Pw water partial vapour pressure at temperature T (Pa) Q0 brine ﬂow rate exiting the nozzle (m3 /s) Re Reynolds number S salinity in terms of NaCl (%) Sc Schimdt number Sh Sherwood number Sol NaCl salt solubility in the water (g NaCl/100 g H2 O) T brine temperature (◦ C) Tair air temperature (◦ C) Tdewpoint dew-point temperature (◦ C) mean ﬁlm temperature (◦ C) Tﬁlm Adrop A0 Cd Cp Cpair d d¯

Tsky Tsp U U0 U* Vbrine Vdrop VNaCl VR Vsp W WR x X Xm Xsat Xsat Xsp y yR z z0

sky temperature (◦ C) mean spray temperature (◦ C) drop velocity relative to the ground (m/s) velocity of the drop exiting the nozzle (m/s) friction velocity (m/s) volume of the brine solution in the drop (m3 ) total drop volume (m3 ) total volume of salt precipitated in the drop (m3 ) relative velocity of the drop with respect to the wind (m/s) total spray volume-by-volume unit of entering brine (m3 /m3 ) wind velocity relative to the ground (m/s) wind velocity at the reference height (m/s) position of the drop along the x-direction (m) salt concentration in the drop brine solution (kg/m3 ) mole fraction of the water salt concentration saturation in the drop brine solution (kg/m3 ) concentration increment above its saturation value (kg/m3 ) mean spray salt brine concentration (kg/m3 ) position of the drop along the y-direction (m) reference height (m) position of the drop along the z-direction (m) aerodynamic roughness length (m)

Greek symbols ˛ angle of the brine jet at nozzle exit with respect to the horizontal direction (◦ ) ˇ angle of the brine jet at nozzle exit with respect to the y–z vertical plane (◦ ) angle between wind velocity vector and x-axis in the horizontal plane (◦ ) ε surface emissivity Tevap water latent heat of vapourization (J/kg) air air viscosity (N/(s m2 )) air density (kg/m3 ) air brine density of brine solution in the drop (kg/m3 ) drop apparent density (kg/m3 ) drop NaCl solid salt density (kg/m3 ) sat brine saturation density in the drop (kg/m3 ) vapour sat,T water vapour density in the saturated state at drop temperature (kg/m3 ) vapour sat,Tair water vapour density in the saturated state at air temperature (kg/m3 ) 0 auxiliary variable used in brine density calculations (kg/m3 ) Stefan–Boltzmann constant air relative humidity

water losses in sprinkler irrigation of a canopy plant under ﬁeld conditions. Their procedure combines equations governing water droplet evaporation, based on the heat and mass transfer analogy, and droplet ballistics with a plant-environment energy model. Kincaid and Langley (1989) model, which was evaluated with laboratory data, allows predicting evaporation and temperature changes in water drops travelling through air. It uses combined sensible heat transfer and diffusion theory in an energy balance to simultaneously calculate evaporation as droplet temperature approaches wet bulb temperature of the air. Mokeba, Salt, Lee, and Ford (1997) proposed a model that combines both ballistic and random-walk models to describe the three-dimensional dynamics

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of spray droplets, under various weather conditions. Lima et al. (2002) presented a three-dimensional model of a downwardspraying rainfall simulator, which accounts for the kinetic energy of the falling water droplets in the presence of wind. Lorenzini (2004) proposed and validated an easy-to-use mathematical model for irrigation sprinkler droplet ballistics and analyzed the effect of air friction in droplet evaporation during its aerial path. Singlesprinkler wind-distorted distribution patterns were analyzed and computed by some authors, through drop trajectory computations and the use of an empirical drag coefﬁcient (Fukui, Nakanishi, & Okamura, 1980; Seginer, Nir, & von Bernuth, 1991; Vories, von Bernuth, & Mickelson, 1987). With the same goal, Okamura and Nakanishi (1969) followed an approach based on momentum and drag coefﬁcients. Another work to consider is the one developed by Carrión, Tarjuelo, and Montero (2001), which allows obtaining wind distorted water distributions, as well as, evaporation and drift losses, through the use of ballistic theory and a new formulation for the air drag coefﬁcient. Their model is implemented in SIRIAS software, which presents the results in a user-friendly environment. As mentioned above some models follow the statistical approach, as for instance the ones given by Tarjuelo, Ortega, Montero, and Juan (2000) and Playán et al. (2005), which allow predicting water evaporation and drift loss in irrigation systems through a multi-parameter experimental investigation. Gil and Sinfort (2005) presented a review about spray drift and pesticide deposit. Some example drift models and studies are the ones of Baetens et al. (2007), Holterman, Van De Zande, Porskamp, and Huijsmans (1997), Miller and Hadﬁeld (1989), and Thompson and Ley (1983). Spray drift during ﬁeld spraying can be inﬂuenced by several factors, from technical to environmental ones. The most relevant factors for the spray system to be used are: 1. 2. 3. 4. 5. 6.

Droplet diameter; Operating pressure; Nozzle type; Nozzle height; Wind velocity; Air humidity.

The droplet size is one important factor concerning spray drift. Smaller drops lead to higher drift values. Drops diameters are inﬂuenced through the choice of nozzle type and spray pressure. Higher pressures lead to inferior drops sizes, thus to higher drift. Raising the nozzle height also increases drift potential because the travelling time of the small droplets increases. The wind speed at the time of spraying is another important factor that inﬂuences spray drift. Higher wind values correspond to an increase of spray drift. Lower air humidity values leads to an augment of the spray drift potential since it increases the evaporation from the droplets during their ﬂight, and therefore leads to a reduction of droplet size (Asman, Jorgensen, & Jensen, 2003). In order to build a complete mathematical model of a spray system it is necessary to predict the water evaporation rate during spraying. One of the most signiﬁcant studies in water evaporation calculations was published by Ranz and Marshall (1952a, 1952b). They studied the evaporation rates of pure water drops and obtained useful independent correlations of the heat and mass transfer rates. They also analyzed evaporation rates of water drops containing dissolved and suspended solids. The evaporation of droplets containing dissolved solids was found to be lower than for that of pure drops, since the presence of the solids reduces the liquid normal vapour pressure. This effect was not observed in the suspended solids droplets, in which evaporation rates correspond to the pure liquids values.

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Most of the studies for the evaporation rates of water droplets with dissolved or suspended solids available in the literature refer to particle drying purposes. In a spray drying process, liquid drops containing solids exit the spray nozzle at low velocities into a very hot surrounding air. Typically, the evaporation and drying process can be subdivided into two periods. The ﬁrst period is characterized by evaporation from a free liquid surface, since movement within the drop is rapid enough to maintain it. Evaporation occurs at a relatively constant rate, resulting in shrinkage of the droplet diameter. In this so-called constant rate period, drying rate is generally controlled by the velocity, ﬂow pattern, temperature and humidity of dry air. Through time, droplet temperature will approach air wetbulb temperature, for drops with suspended solids, and a higher equilibrium temperature, for drops with dissolved solids, due to the observed reduction in the vapour pressure. At this stage the rate of mass transfer balances the rate of heat transfer. During the second drying period, evaporation occurs from or through a solid structure. When critical moisture content is reached, and water supply from the interior to the surface of the drop becomes inadequate, dry spots start to appear. In this falling rate period a dry solid porous crust surrounding a wet core is present, and an additional resistance to both heat and mass transfer arises due to the porous solid layer (Dalmaz, 2005; Perry & Green, 1997). Charlesworth and Marshall (1960) presented an initial study on the evaporation from single drops containing dissolved solids, with the purpose of predicting its drying rates for application in spray driers. In their study drops were suspended over a hot air stream as its drying behaviour was observed and its weight changes were measured. Nesic and Vodnik (1991) studied the evaporation of droplets containing dissolved or dispersed solids. They considered a ﬁve stage concept in the droplet evaporation during the drying process: (1) initial heating and evaporation; (2) quasi-equilibrium evaporation; (3) crust formation and growth; (4) boiling; and (5) porous particle drying. The ﬁrst two stages are often called low-temperature period or constant drying rate period. The other three periods correspond to the high-temperature period or declining drying rate period. Elperin and Krasovitov (1995) analyzed the evaporation of a liquid droplet containing small solids particles (slurry droplet). Farid (2003) presented a model in which it was assumed that droplet undergoes four stages of heating. Dalmaz, Ozbelge, Eraslan, and Uludag (2007) presented a radial distributed model based on a receding evaporating front approach associated with a simultaneously heat and mass transfer. Mezhericher, Levy, and Borde (2007) presents a more realistic model for drying of liquid droplets with dissolved and insoluble solids, which tries to overcome some shortcomings of existing published models. The main goal of this study is to build a model of a heated brine spray system, in order to predict its behaviour through dynamic simulation and to optimize its performance. In this system a concentrated brine solution, with initial temperatures from 65 to 85 ◦ C, is sprayed into the surrounding air (temperatures around 5–30 ◦ C), subject to wind conditions. This spray system includes a set of nozzles that are distributed along the salt (NaCl) recrystallization ponds integrated with the cogeneration system located in Pombal, Portugal (Moita, Matos, Fernandes, Nunes, & Prior, 2005). Its objective is to help increasing the water evaporation, and therefore the salt production, as well as, cooling the brine inside the ponds in order to maintain the water temperatures in the associated cogeneration system within its operational interval. So, in this work it is presented a new three-dimensional dynamic mathematical model of a spray system, which allows, through simulation, to predict its behaviour and to optimize its performance by studying the effect of some variables. The developed model applies to spraying of concentrated heated brine solutions into the surrounding air, at ambient temperatures and under wind conditions. It must be pointed out

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that the modelling of this particular system was not found in the literature review, since most papers referring to spray systems modelling show features at either different conditions (hot air, low velocities of drops, drift issues, etc.) or different solutions (one pure component—water, or suspended solids, for instance). As mentioned previously, most papers refer to spray system applications areas such as sprinkler irrigation, spray drying, ﬁre extinguishment, etc., which do not represent either the characteristics of the mentioned modelled system or its atmospheric and operating conditions or nozzles types. In the next section a dynamic model of the spray system is presented through a single drop model that allows determining the droplet movement and behaviour after exiting the nozzle and through its ﬂight; and the full-cone spray-nozzle model that considers a set of random statistical deﬁned droplets. In Section 3 some results of the dynamic simulation and the validation of the spray system model are shown. In Section 4 the conclusions and future work are presented. 2. Dynamic model of the spray system A three-dimensional dynamic mathematical model for the heated brine spray system was developed, to predict by simulation its behaviour. Two models were developed: one for a single drop and one for the spray system. The single drop model, which is based on the ballistics theory and includes material and energy balances, allows calculating the trajectory and velocity of the drop once catapulted from a nozzle, as well as its temperature, salt concentration, and volume during the ﬂight. The spray system model accounts for the effect of all the drops hits, released from a single full-cone spray-nozzle, by considering a set of random deﬁned drops. 2.1. Single drop model

Fig. 1. Three-dimensional scheme of the brine drop ﬂight through the air in wind conditions.

Fig. 1 shows the forces acting on the drop: (1) gravity force, in the vertical direction; (2) drag force, which opposes the relative movement of the drop in the air, and therefore in wind conditions is not tangential to the brine jet segment; and (3) buoyancy force, in the vertical direction. So, by considering these acting forces in the drop, and applying Newton’s second law of motion:

∂(mdrop U)

F =

it is possible to obtain a set of differential equations that allow describing the path and velocity of the drop exiting the spray nozzle: d(mdrop Ux ) dt

The single drop model considers a dilute dispersed phase ﬂow, since the particle motion is controlled by the ﬂuid forces, rather than by particle collisions, as deﬁned in Crowe et al. (1998). It was not accounted for particle–particle interactions. It was also considered a one-way coupled ﬂow, that is, the ﬂow of one phase affects the other, while there is no reverse effect (Crowe et al., 1998). In the developed model the presence of droplets does not affect the gas ﬂow ﬁeld (air), while the gas ﬂow ﬁeld is responsible for the change in particle velocity, temperature, mass, and concentration, for instance. It was also considered mass and energy coupling. In the modelling of a single drop exiting from the nozzle of the spray to the air, in wind conditions, it was considered a spherical drop shape during all the fall. This is consistent with the photographic studies of Okamura and Nakanishi (1969), in which it is concluded that this is a reasonable assumption. 2.1.1. Ballistics theory In Fig. 1 it is represented a three-dimensional scheme of the drop ﬂight through the air, after exiting the spray nozzle. In a non-wind situation a two-dimension analysis should be sufﬁcient, since the drop trajectory would be in the vertical plane. However, due to the presence of the wind, which is considered only in the horizontal plane, a three-dimensional model is required. The wind inﬂuences the drop trajectory and changes its velocity, which is calculated through (Seginer et al., 1991): −W V R = U

(1)

where V R is the relative velocity of the drop with respect to the is the drop velocity relative to the ground and W is the wind, U wind velocity also relative to the ground. The drop relative velocity V R is not tangential to the brine jet segment, contrarily to the drop ground velocity U.

(2)

∂t

=−

Cd d2 air (Ux − Wx )VR 8

(3)

(iii)

(i)

d(mdrop Uy ) dt

= −mdrop g − (ii)

(i)

Cd d2 air (Uy − Wy )VR + mdrop g air 8 drop (iii)

(iv)

(4)

d(mdrop Uz ) dt

=−

Cd d2 air (Uz − Wz )VR 8

(5)

(iii)

(i)

where the term (i) relates to the acceleration of the drop, term (ii) to the gravity force, term (iii) to the drag force, and term (iv) to the buoyancy force. mdrop and drop are the drop mass and apparent density, respectively, d the drop diameter, and g the gravitational acceleration. The drop velocity components Ux , Uy and Uz are determined through the equations of drop motion: dx = Ux dt

(6)

dy = Uy dt

(7)

dz = Uz dt

(8)

where x, y and z give the position of the drop along the x-, y- and z-direction. The absolute value of the relative drop velocity VR is calculated by:

VR =

(Ux − Wx )2 + (Uy − Wy )2 + (Uz − Wz )2

(9)

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with the wind velocity components Wx , Wy and Wz determined through:

Wx = W cos Wy = 0 Wz = W sin

(10)

180

(11)

(12)

180

where is the angle between the wind velocity vector and the xaxis, in the horizontal plane (see Fig. 1). To calculate the absolute wind velocity value W it was assumed a vertical logarithmic wind proﬁle above the ground surface (Lima et al., 2002; Stull, 2000): W=

U∗ K

Ln

y z0

for

y > z0

(13)

with the friction velocity U* given by: U∗ =

KWR Ln(yR /z0 )

(14)

K is the Von Karman constant equal to 0.4; y the height above the ground along the y-direction, perpendicular to the surface; and WR is the wind velocity at the reference height: yR = 10 m (standard measuring height at meteorological stations). For the aerodynamic roughness length (z0 ), which is chosen so that wind velocity is zero at that height, a value of 0.005 was used. As a simpliﬁcation of the jet break-up process, it was considered that the brine jet is disintegrated at the nozzle exit into individual drops, with different sizes, moving independently in the air. So, for each drop the drag coefﬁcient Cd is considered to be a function of the Reynolds number of a spherical drop, as well as independent of the spray height, discharge angle of the jet, wind velocity, nozzle diameter, and other factors (Carrión et al., 2001). The drag coefﬁcient assumed is the one given by Perry and Green (1997): (a) for Re < 0.1: Cd =

24 Re

(15)

Cd =

24 Re

(1 + 0.14Re0.7 )

(16)

(c) for 1000 ≤ Re < 3.5 × 105 : Cd = 0.445

VR dair air

(20)

with Vbrine as the volume of the brine solution in the drop and brine as the brine density, which is dependent of salt concentration (X) and temperature (T) values according to Eqs. (A3) and (A4). The water evaporation rate (Mevap ) is determined by considering a convective mass transfer of the water vapour from the droplet surface to the surrounding air, and by assuming thermodynamic equilibrium at the interface between the gas and the liquid phase and an ideal gas behaviour for the water vapour in the free stream (Incropera & DeWitt, 2001): Mevap = hm Adrop (vapour sat,T Xm − vapour sat,Tair )

(21)

where is the air relative humidity; Adrop is the drop surface area (Adrop = d2 ); and where vapour sat,T and vapour sat,Tair are the water vapour densities in the saturated state, at the drop surface and air temperature, T and Tair , respectively, which are calculated through Eqs. (A5) and (A6). Due to the presence of the dissolved salt (NaCl) the water evaporation rate in the brine solution is lower than in the pure water liquid. Therefore, to account this effect, it is introduced the corrective term Xm , which is the mole fraction of the water, given by Sartori (1991): Xm =

1 1 + 0.621(S/(100 − S))

(22)

where S is the salinity in terms of NaCl, which relates to salt concentration: S = 100

X birne

(23)

The average convective mass transfer coefﬁcient hm is determined from the widely used correlation for the free falling of liquid drops given by Ranz and Marshall (1952a, 1952b): (24)

This equation was found to be representative of the experimental data and it has, according to Knudsen and Katz (1958) and Ranz (1952), the following parameters range: (1) 1 < Re < 70,000; (2) 0.6 < Sc < 400.

(18)

The air density (air ) and viscosity (air ) are determined through correlations given by Eqs. (A1) and (A2) in Appendix A, respectively. 2.1.2. Material balances The material balances were performed considering the ﬂight of the spherical drop after been catapulted of the spray nozzle, and a ﬁnite time interval t, in which there is water evaporation and salt precipitation. Assuming that both brine density and salt concentration in the solution do not change within the drop diameter, no radial proﬁles exist. So, applying the law of mass conservation for the brine solution contained in the drop over that time interval, and after mathematical manipulation, results into the brine material balance equation (Bequette, 1998): dmbrine = −Mevap − MNaClpp dt

mbrine = brine Vbrine

(17)

with Reynolds number Re determined through: Re =

where the total mass of the brine in the drop is determined by:

Sh = 2 + 0.6Re1/2 Sc 1/3

(b) for 0.1 ≤ Re < 1000:

1327

(19)

Sh is the Sherwood number (Sh = hm d/Dab ) and Sc the Schimdt number (Sc = air /(air Dab )). The mass diffusivity of water vapour through air, Dab , is calculated by Eq. (A7). So, as it can be seen from Eq. (21), the water evaporation rate depends on atmospheric conditions (air temperature, and humidity; and wind velocity), brine temperature and salt concentration, as well as, drop dimension and velocity. The salt material balance is determined by applying the law of mass conservation for the salt presented in the brine solution contained in the drop over the same time interval considered earlier (Eqs. (25)–(28)). It is assumed that precipitation only occurs when the concentration of the brine solution is Xsat above its saturation value (X − Xsat ≥ Xsat ). The salt precipitation rate MNaClpp is determined through the material balance equation (Eq. (26)) and the salt concentration value is ﬁxed (Eq. (25)). If the solution is not supersaturated, X will be computed by the balance equation (Eq. (28)), with MNaClpp = 0. It was assumed a value of Xsat = 0.05 kg/m3 .

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the components in their state of aggregation at 25 ◦ C (that is, liquid water and solid NaCl). Thus, the enthalpy of the liquid brine drop with a volume (Vbrine ), salt concentration (X), temperature (T) and brine density (brine ) is determined through (Coulson & Richardson, 1989; Houghen, Watson, & Ragatz, 1972): Hbrine = Vbrine [(brine − X)Cpwater (T − 25) + XCpNaCl (T − 25) ◦

25 C + XHdiss ]

The brine saturation concentration Xsat is calculated through Eq. (29): Xsat =

Sol sat Sol + 100

35.549 − 0.23125T 1 − 0.0069163T

Eevap = Mevap [Tevap + Cpwater (T − 25)]

(30)

The total solid salt mass obtained in the drop (mNaClpp,total ) is determined by: d(mNaClpp,total ) dt

= MNaClpp

(31)

and relates to the total volume of salt precipitated in the drop (VNaCl ) by: mNaClpp,total = VNaCl NaCl

(32)

where a solid salt density (NaCl ) value of 2170 kg/m3 was used. The total drop volume (Vdrop ) and mass (mdrop ) are given by Eqs. (33) and (34), respectively: d3 Vdrop = = VNaCl + Vbrine 6 mdrop = mbrine + mNaClpp,total

(33) (34)

with drop brine and salt masses determined through its respective mass balances equations (Eqs. (19) and (31)), and volumes through Eqs. (20) and (32). The drop apparent density (drop ) takes into account the existing brine solution and the precipitated salt in the drop, and it is calculated by: drop =

mdrop Vdrop

(35)

2.1.3. Energy balance As before, it was considered the spherical drop during its ﬂight and the same ﬁnite time interval t, in which there is water evaporation and salt precipitation. However, in this case, it is also accounted for the energy losses due to convection and radiation, as well as the solar energy, which is absorbed through the brine drop. The resulting energy balance is given by: dHbrine = −Eevap − Econv − Erad + Esolar + ENaClpp dt

Cpwater and CpNaCl are mean heat capacity values calculated using the expressions given by Perry and Green (1997), and are equal to 4.189 × 103 and 0.8712 × 103 J/(kg ◦ C), respectively. The heat of 25 ◦ C is determined through Eq. (A8). dissolution Hdiss Evap is the enthalpy of a vapour water stream due to evaporation, and according to the enthalpy reference state deﬁned, is calculated by

(29)

where the salt solubility in the water Sol is given by Eq. (30) (Langer & Offermann, 1982), and the saturation density sat by Eqs. (A3) and (A4), using the Xsat value (Eq. (29)). Sol =

(37)

(36)

It was assumed a uniform temperature distribution across the drop volume, neglecting any gradient temperature proﬁle, since its Biot number is inferior to 0.1, and therefore a lumped model can be used (Incropera & DeWitt, 2001). Hbrine is the total enthalpy of the brine solution contained inside the drop. The enthalpy reference state considered corresponds to

(38)

with the water evaporation rate Mevap determined by Eq. (21) and the water latent heat of vapourization Tevap by Eq. (A9). Due to the movement of the heated drop in the air there are energy losses, caused by forced convection, which are determined through (Incropera & DeWitt, 2001): Econv = hc Adrop [T − Tair ]

(39)

The average convective heat transfer coefﬁcient hc is, once again, determined from the known correlation given by Ranz and Marshall (1952a, 1952b): Nu = 2 + 0.6Re1/2 Pr 1/3

(40)

with the following parameters range (Knudsen & Katz, 1958; Ranz, 1952): (1) 1 < Re < 70,000; (2) 0.6 < Pr < 400. Nu is the Nusselt number (Nu = hc d/kair ) and Pr the Prandlt number (Pr = Cpair air /kair ). The air conductivity (kair ) and air heat capacity (Cpair ) are determined though Eqs. (A10) and (A11), respectively. Therefore, the calculated convection energy losses, given by Eq. (39), vary with atmospheric conditions (air temperature and wind velocity), brine temperature and drop dimension and velocity. The radiation energy losses through the brine drop surface (Erad ) depends on the sky and brine temperatures (Incropera & DeWitt, 2001): Erad = Adrop ε[(T + 273.15)4 − (Tsky + 273.15)4 ]

(41)

where ε is the surface emissivity (equal to 0.95), is the Stefan–Boltzmann constant (5.67 × 10−8 W/(m2 K4 )) and Tsky is the sky temperature calculated by Eq. (42) (Sartori, 1996):

Tsky = [Tair + 273.15] 0.8 +

Tdewpoint 250

1/4 − 273.15

(42)

The dew-point temperature (Tdewpoint ) is obtained through the psychrometric charts presented by Perry and Green (1997) using Eq. Tdewpoint Tair are the water vapour partial presand Pw (43), where Pw sure at dew-point and dry air temperature, respectively, and are determined by Eq. (A12), by replacing those values. Tdewpoint

Pw

Tair = Pw

(43)

R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

The enthalpy of the salt (NaCl solid) precipitated stream in the drop (ENaClpp ), is computed by: ENaClpp = CpNaCl (T − 25)MNaClpp

(44)

The extraterrestrial solar radiation, at the top of the atmosphere, depends on the geographic coordinates as well as on the time of the day, month and year. The solar radiation that reaches the earth’s surface (G) is lower due to absorption and scattering by the atmospheric constituents, and is a sum of the direct and diffuse contributions. Part of this radiation that reaches the brine drop surface is reﬂected, part is absorbed and the rest passes through by transmission. The solar energy value considered in the energy balance equation (Eq. (36)), Esolar , is the absorbed radiation fraction, which is calculated by (Incropera & DeWitt, 2001): Esolar = fabs GAdrop

(45)

where fabs corresponds to the total hemispherical absorptivity and it was considered equal to 0.9 (Sartori, 1996).

It was also necessary to deﬁne initial values for the drop diameter (d), brine temperature (T) and salt concentration (X), as well as for the total salt mass precipitated in the drop (mNaClpp,total ). 2.2. Spray system model Once determined the mathematical equations that describe the behaviour of a single drop leaving the nozzle, it is possible to develop the dynamic model of the whole spray system, which accounts for all the drops leaving from the full-cone spray-nozzle, by considering a set of random deﬁned drops. For that purpose it were considered statistical distributions for the drops diameters (d) and for the initial angles of the exiting brine jet (˛, ˇ). For the diameters it was considered the Rosin-Rammler distribution, which is usually used to describe particle size distributions, with its cumulative function given by (Djamarani & Clark, 1997; Kuo, 1986):

m d

f (d) = 1 − exp − 2.1.4. Initial conditions As the model includes ﬁrst order differential equations with time, it must be deﬁned initial values of the state variables, that is, at time zero. Initial values for the drop position (x, y, and z), which corresponds to the nozzle localization, are given by: x|t=0 = 0

(46)

y|t=0 = h0

(47)

z|t=0 = 0

(48)

where h0 is the nozzle height with respect to ground level. The initial drop velocity x–y–z components values are determined by: Ux |t=0 = U0 cos Uy |t=0 = U0 sin Uz |t=0 = U0 cos

˛ 180

sin

ˇ 180

(49)

˛

(50)

180

˛ 180

cos

ˇ 180

(51)

where ˛ and ˇ are the angles of the brine jet, leaving the nozzle at an absolute velocity value of U0 , with respect to the horizontal direction and to the y–z vertical plane, respectively (see Fig. 1). Assuming that there are no velocity losses at the exit of the nozzle, the drop velocity value U0 can be calculated using the values of the brine ﬂow rate exiting the nozzle (Q0 ) and of the oriﬁce area (A0 ): U0 =

Q0 A0

(52)

The oriﬁce area relates to its respective diameter through Eq. (53). The exiting brine ﬂow rate depends on the liquid stream differential pressure P and is given by Eq. (54) (Bete website, 2008). A0 = Q0 =

d02

d¯

√ Kbrine P

(54)

6 × 104

Kbrine is a characteristic brine spray constant that is a corrected term for its respective water value Kwater (Bete website, 2008), and is determined by:

Kbrine = Kwater

water brine

(56)

where d¯ (mean diameter) and m (a measure of the spread particle size distribution) are determined through measured values obtained from the spray nozzle supplier (Bete website, 2008), for the used spray nozzle. The values of the angles ˛ and ˇ are calculated by a uniform distribution. This modelling strategy, developed and followed here, is analogous to the Lagrangian approach (Crowe et al., 1998). In this approach the velocity, mass and temperature history of each particle (or representative particle) in the cloud are calculated. Through its discrete element method, the motion and position (as well as other properties) of individual particles, or representative particles, are tracked with time. Since is not computationally feasible to track each and every particle (there are too many particles), a smaller number of computational particles (sample) are chosen to represent the actual particles. One computational particle is regarded as representative of a parcel of particles. It is assumed that parcel of particles moves through the ﬁeld with the same velocity and temperature, etc., as a single particle (physical particle). The motion of each parcel over one time interval is obtained by integrating the particle motion equation. At every time step, the properties of the droplet cloud can be determined by summing over all the particles in a computational volume (Crowe et al., 1998). Moreover, some global variables for the entire spray system can be determined by accounting the results for each drop obtained with the dynamic model developed and presented in Section 2.1. For that purpose, consider a set of n drops, deﬁned according to the mentioned statistical distributions, leaving the spray nozzle and reaching the end of its ﬂight, as it lands in the ground, as illustrated in Fig. 2. So, each drop i of the set, with i = 1–n, have a diameter di , temperature Ti , salt concentration Xi and brine and precipitated salt mass mbrine,i and mNaClpp,total,i , respectively. The total spray volume (Vsp ) obtained at the end of its falling time, by unit of brine volume sent by the nozzle, is determined

(53)

4

1329

(55) Fig. 2. Calculation of the global spray system model output variables.

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R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

through Eq. (57): Vsp =

V drop,i

(57)

Vdrop,i |t=0

The spray total brine mass (Mbrine,sp ) and salt precipitated (MNaClpp,sp ), by volume unit of entering brine, is calculated by Eqs. (58) and (59), respectively: Mbrine,sp =

m brine,i

(58)

Vdrop,i |t=0

mNaClpp,total,i MNaClpp,sp =

Fig. 3. Implementation of the dynamic model in gPROMS 2.3.7.

(59)

Vdrop,i |t=0

The mean spray values for the salt brine concentration Xsp and temperature Tsp , are calculated through salt material and energy balances:

(Xi Vbrine,i ) = Xsp

Vbrine,i

Hbrine,i = Cpwater (Tsp − 25) ×

(60) (mwater,i ) + CpNaCl (Tsp − 25) ◦

25 C (mNaCl,brine,i ) + Hdiss

(mNaCl,brine,i )

(61)

where for each drop i, the mass of water is given by mwater,i = mbrine,i − Vbrine,i Xi , the salt mass in the brine solution by 25 ◦ C through Eq. mNacl,brine,i = Vbrine,i Xi , and its heat of dissolution Hdiss (A8). The total water evaporated by volume unit of entering brine volume, during the spraying time, is: Mevap,sp =

M dt evap,i

(62)

Vdrop,i |t=0

The global evaporative spray efﬁciency is given by:

Eff =

Eevap,i dt

(63)

(Eevap,i + Econv,i + Erad,i ) dt

3. Dynamic simulation and validation of the spray system model The whole spray system was modelled through the generalpurpose modelling, simulation and optimization tool gPROMS 2.37, of the Process System Enterprise Ltd. This software allows to adequately handling process discontinuities, lumped and distributed systems and many different types of operating procedures (gPROMS, 2004).

Fig. 3 is an information ﬂow diagram illustrating the dynamic model structure implemented in this software. It accounts for the developed single drop and spray system models. The global model, that is the Spray System model, contains the necessary equations to determine the global spray variables, as discussed in Section 2.2. It interacts with the calculated single drop variables since it includes n sub-models, labelled Single Drop i model, with i = 1–n, in which are represented all the algebraic and differential equations needed to describe the behaviour of each brine drop exiting the nozzle, as described in Section 2.1. In order to implement and solve the dynamic model of the spray system it is necessary to deﬁne the initial diameter (di ) and the exit spray angles (˛i and ˇi ) of each drop. These values should follow statistical distributions to be randomly generated, as stated before. This is possible through stochastic simulation in gPROMS. In the process section of this software, instead of assigning deterministic values for these variables, special functions are used that return values sampled from the distribution chosen. It was considered for the diameter di the Weilbull (Rosin-Rammler) distribution function, while for the exiting angles ˛i and ˇi a uniform distribution function. The simulation of the set of n drops occurs simultaneously. As the falling time of each drop is different, it is necessary to consider a stopping criterion for the whole simulation time. Each drop ﬂight ends as it lands in the ground, that is, when the drop vertical position yi is equal to zero. So, the stopping criterion variable for each drop Criti is equal to zero nif yi = 0 and equal to one otherwise. Thus, simulation ends when Criti = 0. This ensures that all drops have 1 ﬁnished their ﬂight. Furthermore, the right side of the differential equations, which describe the single drop model, were multiplied by the stopping criterion variable Criti . This way, when Criti = 0 (drop has reached the ground), the derivatives are also zero, and the respective

Table 1 Spray system model analysis and performance statistics. Main input parameters/variables Spray characteristics (s(t)) Drop diameter distribution ˛ and ˇ values Nozzle oriﬁce diameter Kbrine /Kwater Atmospheric conditions (d(t)) Tair , W, , Esolar Operational conditions (u(t)) Operating pressure Nozzle height Initial conditions for each drop i Ti (0), Xi (0), MNaClpp,total,i (0)

Main output variables Spray system Tsp Xsp MNaClpp,sp Mbrine,sp Mevap,sp Eff

Typical performance statistics Number of drops

20

Number of variables Differential Algebraic

300 2,319

1,200 9,159

Number of equations

2,619

10,359

392

7,718

gPROMS execution time (s)a

80

Single drop i Position and velocity di Ti and Xi mbrine,i and mNaClpp,total,i

s(t): parameters that correspond to a spray type; d(t): input variables that correspond to external disturbances; u(t): input variables that can be manipulated to optimise the system performance. a Pentium 4, 3.4 GHz, 1.00 GB RAM.

R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

differential variables will remain unchanged through the simulation. This guaranties that the values of all variables at the end of simulation time correspond to its respective falling values. All the operational and atmospheric conditions, as well as spray characteristics, are accounted for in the simulations. The total number of model variables depends on the number of drops considered (n). In Table 1 a model analysis and some performance statistics are presented. From this table is possible to verify that an increase of the amount of drops requires a greater computational effort. 3.1. Sensitivity analysis In this section a sensitivity analysis of some model parameters and choices is presented, by considering the spraying of heated brine solution, in wind conditions, with temperatures around 70 ◦ C and spray ﬂow rates between 50 and 70 m3 /h, at an initial height between 1.2 and 2.6 m. Initial angles ˛ and ˇ are within 15–75◦ , and two different spiral full-cone spray nozzles from BETE were considered (Bete website, 2008): TF88: Flow rates = 27–171 m3 /h, d50 (1 bar) = 2.3 mm, Kwater = 638, dfree pass = 11.1 mm. TF112: Flow rates = 50–313 m3 /h, d50 (1 bar) = 2.4 mm, Kwater = 1170, dfree pass = 14.3 mm. To determine the average convective mass and heat transfer coefﬁcients (hm and hc ), through Eqs. (24) and (40), respectively, the physical properties of the air should be evaluated at air temperature, according to Incropera and DeWitt (2001). However, if a mean ﬁlm temperature is used instead, that is, if Tﬁlm = (Ti + Tair )/2 is used, it will be obtained an increase in total water evaporated Mevap,sp of around 3 and 5%, for TF88 and TF112 nozzles, respectively, and a decrease in spray temperature Tsp of around 2% for both nozzles. The energy losses due to radiation to the open air are calculated through Eq. (41), in which the sky temperature is used. If, as a simpliﬁcation, the air temperature value is used instead, differences in water evaporation and temperatures will be lower than 0.11%, since radiation energy loss is very small (global percentage contribution is inferior to 1%). Thus, if it is necessary to minimize computational effort, and to simplify, it can be used Tair instead. The solar contribution is very small, and therefore considering that all solar energy is absorbed, that is fabs = 1 instead of a 0.9 value, leads to differences in Mevap,sp and Tsp inferiors to 0.3% (see Eq. (45)). It was also analyzed the inﬂuence of the drag coefﬁcient (Eqs. (15)–(17)). For that purpose it were considered the expressions given by Perry and Green (1997), Carrión et al. (2001), Lorenzini and Wrachien (2004), and Li and Chow (2008). Differences in Mevap,sp and Tsp values are inferior to 0.3%, while for the maximum travel distance it goes up to 8%. To calculate the wind velocity it was assumed a vertical logarithmic proﬁle (Eq. (13)). An sensitivity analysis for the roughness length (Z0 equal to 0.0002 and 0.005) and standard measuring height (YR from 4 to 10) leads to deviations inferior to 0.1% in the total spray evaporation and temperatures values. Table 2 presents a summary of the sensitivity analysis results.

1331

Table 2 Summary of the results of the model parameters and choices sensitivity analysis. Model parameter/choice

Inﬂuence on the spray system

Tﬁlm instead of Tair in Eqs. (24) and (40)

↑Mevap,sp ∼ 3% (TF88) and 5% (TF112) ↓Tsp ∼ 2% (TF88 and TF112)

Tair instead of Tsky in Eq. (41)

Mevap,sp and Tsp < 0.11%

fabs = 1 instead fabs = 0.9 in Eq. (45)

Mevap,sp and Tsp < 0.3%

Different Cd values in Eqs. (15)–(17)

Mevap,sp and Tsp < 0.3% Maximum travel distance < 8%

Z0 (0.0002 and 0.005) and YR (4–10) in Eq. (13)

Mevap,sp and Tsp < 0.1%

Table 3 Literature sprinkler data set used for the comparison analysis.

Flow rate leaving the nozzle (m3 /s) Nozzle diameter (mm) Angle of the exiting jet ˛ (◦ )a Nozzle height (m) Air temperature (◦ C) Wind (m/s) a

Edling (1985) data

Thompson et al. (1993) data

1.4 × 10−4 3.96 10 1.22 2.44 29.4 0

5.5 × 10−4 4.76 25 4.5 38 0

The angle ˇ is assumed to be equal to 0.

3.2.1. Water drops simulations In this section two types of simulations were considered: one to validate the drop trajectory predictions (through ballistic theory), and another to validate the evaporation model. 3.2.1.1. Trajectory simulations. With the purpose of evaluating how reliable the model trajectory prediction is, it was considered data retrieved from Lorenzini (2004), which uses in its simulations original data from Edling (1985) and from Thompson et al. (1993), for no wind situations. From all cases presented only the ones indicated in Table 3 were considered, due to its similarity with our spraying conditions. In order to perform the required simulations the model was adapted to consider water instead of a brine solution. In Fig. 4 the travel distances predicted by our model are presented using as an input the mentioned literature data given in Table 3, against Lorenzini (2004), Thompson et al. (1993) and Edling (1985) results, for drop diameters from 0.3 to 5.1 mm.

3.2. Model validation In order to validate the developed model, it were compared its predictions, obtained through dynamic simulation, with data retrieved from literature referring to water drops, as well as measured values in the Portuguese industry referring to brine solution spraying.

Fig. 4. Travel distance for water drops obtained with our model against Lorenzini (2004), Thompson et al. (1993) and Edling (1985) results, for the data shown in Table 2.

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R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

Table 4 Total mean deviations of the travel distances values obtained by this work, by Lorenzini (2004), by Thompson et al. (1993) and Edling (1985).

Lorenzini Thompson/Edling (%) This work Thompson/Edling (%) This work Lorenzini (%)

Edling (1985) data

Thompson et al. (1993) data

h0 = 1.22 m

h0 = 2.44 m

h0 = 4.5 m

5.3 4.1 4.4

4.0 4.2 5.3

26.6 11.7 19.7

As it can be seen in Fig. 4, the model shows good agreement, both qualitatively and quantitatively, with the other author’s values. Using Thompson data, for drop diameters inferior to 2 mm, model predictions are more similar to his results, however for higher drop diameters are closest to Lorenzini values. The same is veriﬁed when Edling data is used, that is, the predicted travel distances are closest to Lorenzini results for higher drops diameters, and to Edling results otherwise. A higher deviation is obtained with the largest drop diameter, when considering a nozzle height of 2.44 m. Table 4 presents the total mean deviations of the travel distances values accounting for all drops diameters. Globally, the developed model predictions are closest to Edling (1985) and Thompson et al. (1993) results. 3.2.1.2. Evaporation simulations. In order to validate the presented evaporation model part, water evaporation rates of individual water droplets measured by Kincaid (1989), under different air temperature, humidity and velocity conditions were compared against the developed model predictions. These measurements were made in a small wind tunnel, in which the drop is suspended and where it is possible to control and measure air temperature, humidity and velocity. Kincaid (1989) used a volumetric technique that allows obtaining the initial and ﬁnal drop volume, with evaporating times of 10–120 s, and therefore allowing to calculate the mentioned water evaporation rates. To perform these comparisons it was necessary, once again, to adapt the developed model to consider water instead of the brine solution. In this case it was also required to impose a constant relative drop velocity VR equal to air velocity, instead of calculating it through Eq. (9), to reproduce the experiments conditions. For each drop, simulation time is given by the measured evaporating time, instead of being obtained through ballistic theory. Fig. 5 presents the water loss rates predicted through simulation by our model against measured values by Kincaid (1989), for a set of selected data corresponding to drop diameters from 0.7 to 1.6 mm, and to different atmospheric conditions. It was considered

a set with high-air humidity values ( ≥ 75%): (a) to (b) values; a medium air humidity set ( ≈ 45%): (c) values; and a low-humidity set ( ≤ 35%): (d) to (h) values. As shown in Fig. 5, model predictions are in reasonable agreement with the mentioned measured values (mean deviation is lower than 7%). It must be noted that experimental measuring error increases as the drop diameter is reduced, as mentioned by Kincaid (1989). Simulations also allowed verifying that higher evaporation rates are obtained for lower air humidity values and for smaller drop diameters. In all simulations, it was assumed an initial drop temperature equal to 15 ◦ C, which changes through simulation time and approaches its respective wet bulb temperature value (Twet-bulb ), as evaporation takes place. The droplet size and relative velocity inﬂuence the time required to reach the wet bulb temperature. Drop size has a stronger effect than drop velocity. Larger drops take a longer time, while drops with higher velocities require less time. These observations are in agreement with the ones concluded by Kincaid and Langley (1989). For all the simulations results shown in Fig. 5 (d < 1.6 mm and VR ≤ 3 m/s), it may take up to 8 s to approach the wet bulb temperature. 3.2.2. Brine drops simulations In the previous section, the developed model was validated by comparison of model predictions, obtained through dynamic simulation, against data retrieved from the literature referring to water drops. In this section the comparison will account for brine spraying measurements, performed in a Portuguese industry. So, for this purpose, it was used a nozzle from the BETE company (reference TF112), for the spraying of a concentrated heated brine solution. The brine solution was characterized at the exit of the nozzle, that is, measured its initial temperature and concentration. Through the use of a recipient placed in the ground in a chosen location within the expected spray wet area, a sample was taken after the spraying time considered and fully characterized (Prior, Rocha, Matos, & Pinho, submitted for publication). This procedure was repeated to considerer different sample locations. In Table 5 are presented the mean atmospheric and operation conditions for the two experiments performed, considering four different locations measures in each one. The nozzle used has a spiral form, and consequently not all the drops have the same initial velocity, since the exiting area is not equal. To account for this in the simulation, an oriﬁce diameter distribution was considered, leading to a set of different values for the initial drop velocity values. The initial drops diameters were calculated through the RosinRammler distribution. Its parameters were determined by ﬁtting into Eq. (56) data obtained from the spray nozzle supplier for the operating pressures (Bete website, 2008). So, d¯ and m values are equal to 3.525 × 10−3 m and 2.023, for experiment 1; and to 3.297 × 10−3 m and 2.023, for experiment 2, respectively. For the initial angles of the exiting brine jet (˛, ˇ) a uniform distribution was used, with angles from 15◦ to 75◦ . In the simulations 40 drops were considered. Table 5 Mean atmospheric and operation conditions for experimental measures (Prior et al., submitted for publication).

Fig. 5. Water loss rates predicted by simulation with this model against measured values by Kincaid (1989), for different atmospheric conditions and drop diameters from 0.7 to 1.6 mm.

Air temperature (◦ C) Air humidity (%) Wind velocity (m/s) Solar energy (kWh/(m2 day)) Nozzle height (m) Pressure (bar) Brine ﬂow rate (m3 /h) Initial brine temperature (◦ C)

Experiment 1

Experiment 2

25.4 28.6 2.3 4 2.6 0.6 50 64

26.4 37 2.0 4 2.6 0.75 56 69

R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

1333

Table 6 Water evaporated and ﬁnal brine temperature predicted and measured in the industry. Experiment 1

Mevap,sp (kg/m3 ) Tsp (◦ C)

Experiment 2

Measured

Predicted

Measured

Predicted

20.9 44.9

21.4 48.8

27.4 44.5

27.6 49.7

Table 7 Operating and atmospheric conditions considered in the simulation. Operating conditions Pressure (bar) Nozzle height (m) Initial temperature (◦ C) Initial concentration (kg/m3 )

Atmospheric conditions 1 1.2 75 280

Air temperature (◦ C) Air humidity (%) Wind velocity (m/s) Solar energy (kWh/(m2 day))

15 50 2 4

Fig. 7. Water evaporation rate for each drop and for the spray system.

In Table 6 are presented the total water evaporated, by unit of volume of the entering brine, as well as the ﬁnal brine temperature, predicted through simulation and obtained by experiments (Prior et al., submitted for publication), which corresponds to the mean values of the four samples considered. The predicted results are in reasonable agreement with the values measured experimentally in the industrial site. The error is inferior to 2.5% and to 12%, for the evaporation rate and temperature values, respectively. 3.3. Dynamic simulation results In this section are presented some results obtained by dynamic simulation for the spray nozzle TF112 at case base conditions. Table 7 show the values of the operating and atmospheric conditions considered in the simulation. Fig. 6 shows the drop vertical position as a function of the travel distance, for four drops with different diameters, initial velocities and spray nozzle angles values. As it can be seen in this ﬁgure, a drop exiting the spray nozzle under a greater angle reaches a higher position along the y-direction, but also an inferior travel distance (drops (a) and (b)). Bigger drops travel a larger distance (drops (b) and (c)). An increase of the drop initial velocity, when exiting the spray nozzle, will lead to a superior falling distance and height (drops (b) and (d)). Figs. 7 and 8 present the evaporation rate and temperature obtained for each drop and for the spray system. It was generated 80 drops, with diameters between 0.4 and 7 mm, through a RosinRammler distribution. The smaller drops have higher evaporation rates, and therefore correspond to lower temperature values. However, due to its reduced volume against the total spray volume, its contribution for the global spray evaporation rate and tempera-

Fig. 6. Drop vertical position as a function of travel distance, for different drop diameters (d), spray nozzle angles (˛) and initial velocities (U0 ).

Fig. 8. Brine temperature for each drop and for the spray system.

ture values is smaller when compared to higher diameter drops. As mentioned before, it was considered a distribution for the spray angles ˛ and ˇ, as well as for the nozzle oriﬁce diameters (and consequently exiting velocities), which leads to different evaporation rates and temperature values, for similar drop diameters. In a falling time of 4.7 s, a maximum travel distance of 25 m was reached, accounting for all drops. For this scenario, the global evaporative spray efﬁciency was 83%. 4. Conclusions and future work A three-dimensional mathematical dynamic model was built to simulate and predict the behaviour of a heated brine spray system. Two models were developed. On one hand, the single drop submodel, which is based on the ballistics theory and includes material and energy balances, that allows the calculation of each drop trajectory and velocity as it exits the nozzle, as well as its temperature, salt concentration and volume. On the other hand, the spray system model that accounts for the full-cone spray-nozzle by considering a set of random deﬁned drops. The spray system model was implemented and simulated in gPROMS 2.3.7, via its advance feature: stochastic simulation. The implemented methodology applied in this software was described in Section 3. Some speciﬁc software modelling strategies were developed: (i) a stopping simulation time criterion, which ensures that all drops have ﬁnished their ﬂight, since the drops have different falling times; (ii) a zero reset to the various derivatives for the stopped drops, guarantying that the values of all variables at the end of its simulation time corresponds to its respective falling values. Additionally, it was presented some model performance statistics and analysis.

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R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

A sensitivity analysis of some model parameters and choices was performed. Model predictions, obtained through dynamic simulation, were compared with retrieved literature data, referring to water drops, in terms of drop trajectories and of evaporation rates. Model validation has also considered experimental data, obtained in the Portuguese industrial plant of Renoeste, referring to its existing heated brine spraying system. It must be stressed out that the simulated results are in reasonable agreement with the literature values and with the experimentally measured ones. The calculated mean deviations values are inferior to 7 and 2.5%, respectively. The heated brine spray system model developed allows the prediction of its behaviour at the range of the operational and atmospheric conditions established. This is advantageous since it allows the improvement of the system performance through the analysis of different working scenarios. Furthermore, it can help in the selection of the more appropriated spray nozzle to use, by analysing through dynamic simulation its characteristics (drop diameter distribution, spray angles, nozzle oriﬁce diameter and drop exiting velocities). It must be noted also that the model developed in this work can be easily adapted in order to take into account other types of systems and sprays. Future work will include an analysis of the inﬂuence of several operating and atmospheric conditions in the spray system. Furthermore, this model will be included with the existing integrated model of the thermal salt recrystallization process (Moita et al., 2005), in order to analyse the spray system effect on the global industrial process.

Water vapour density in saturated conditions

Acknowledgements

To calculate the mass diffusivity of water vapour though air (Dab ), it was used an expression based on the Chapman-Enskog kinetic theory (Welty, Wicks, Wilson, & Rorrer, 2001) that depends on temperature, pressure, components molecular weights and its Lennard-Jones parameters, which are retrieved from Lienhard and Lienhard (2008) and from Welty et al. (2001). Based on these calculated values of Dab , for several temperatures, it was determined a linear correlation for the mass diffusivity value:

The authors gratefully acknowledge ﬁnancial support from Renoeste and from the Portuguese agency FCT—grant SFRH/BDE/15 533/2004. Appendix A. Calculation of physical properties

The air density (air ) and viscosity (air ) are determined through correlations (Eqs. (A1) and (A2)), which are based on the tabulated values given by Lienhard and Lienhard (2008), at several temperatures values. The calculated maximum deviations, for the temperature interval of 6.8–26.8 ◦ C, are 0.1 and 0.8% for air and air , respectively. 2 air = 1.1959 × 10−5 Tair − 4.6097 × 10−3 Tair + 1.2924

(A1)

air = 4.7211 × 10−8 Tair + 1.7158 × 10−5

(A2)

vapour sat,T = 3.0136 × 10−9 T 4 + 1.2140 × 10−7 T 3 + 1.1134 × 10−5 T 2 + 3.2672 × 10−4 T + 4.8735 × 10−3 (A5) (b) for temperatures T > 40 ◦ C: vapour sat,T = 5.1521 × 10−9 T 4 − 2.5053 × 10−7 T 3 + 3.6808 × 10−5 T 2 − 4.9682 × 10−4 T + 1.5079 × 10−2

(A6)

with a deviation value inferior to 0.015% for temperatures values between 4 and 40 ◦ C, for Eq. (A5), and inferior to 0.02% for temperatures between 40 and 100 ◦ C, for Eq. (A6). So, to calculate the water vapour density in saturated conditions at drop surface temperature T (vapour sat,T ) Eqs. (A5) and (A6) are applied, while its value at air temperature Tair (vapour sat,Tair ) only requires using Eq. (A5). Mass diffusivity of water vapour though air

(A7)

The calculated maximum deviation value in a temperature range of 0–40 ◦ C is 0.53%, by comparison to known literature values (Lienhard & Lienhard, 2008; Perry & Green, 1997; Welty et al., 2001). Heat of dissolution of the salt in the water ◦

25 C is determined through a linThe salt heat of dissolution Hdiss ear correlation based on the enthalpies of formation values given by Wagman et al. (1982), for X > 150 kg/m3 : ◦

25 C Hdiss = 1.3712

n

water

nNaCl

+ 19.4063 103

(A8)

where n represents the number of moles of the respective component.

Liquid brine density The brine density is related with salt concentration in the brine solution (X) and temperature (T) values through Eqs. (A3) and (A4). This correlation was obtained using the tabulated values of the density at different temperatures and salt mass fractions (FM ) presented in Perry and Green (1997). The expression was modiﬁed in Mathematica 4.1.0.9 of the Wolfram Research Inc. to eliminate FM (FM = 100X/): brine = 0.5 0 +

(a) for temperatures T ≤ 40 ◦ C:

Dab = 1.6892 × 10−7 Tair + 2.1916 × 10−5

Air density and viscosity

Based on the tabulated values for the water vapour density in the saturated state (vapour sat,T ) retrieved from Harvey (1998) two correlations were calculated:

02 + 400X(7.7780 − 0.0063176T )

0 = 1001.23 − 0.22715T − 0.0020480T 2

Water latent heat of vapourization The water latent heat of vapourization Tevap is determined as a function of the temperature through a linear correlation based on the values given by Daubert (1985): Tevap = [2503.0 − 2.432T ]103

(A9)

(A3)

Air conductivity and heat capacity

(A4)

The air conductivity (kair ) and heat capacity (Cpair ) are determined through correlations (Eqs. (A10) and (A11)), which are based

R.D. Moita et al. / Computers and Chemical Engineering 33 (2009) 1323–1335

on literature values given by Lienhard and Lienhard (2008), at several temperatures. The calculated maximum deviations, for the temperature interval of 6.8–26.8 ◦ C, are 0.33 and 0.04% for kair and Cpair , respectively: kair = 7.1083 × 10−5 Tair + 2.4236 × 10−2

(A10)

2 Cpair = 1.9481 × 10−4 Tair + 2.8976 × 10−2 Tair + 1005.87

(A11)

Water vapour partial pressure The water vapour partial pressure at a temperature T is given by (Perry & Green, 1997):

Pw = exp 73.649 −

7258.2 − 7.3037 Ln[T + 273.15] T + 273.15

+ 4.1653 × 10−6 (T + 273.15)2

(A12)

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