Theoretical Analysis of Saline Diffusion during 2 Sodium Chloride Aqueous Solutions Freezing for 3 Desalination Purposes 4

1 Centro Universitario Tonalá, Universidad de Guadalajara, Av. Nuevo Periférico No. 555 Ejido San José 8 Tatepozco, C.P. 45425, Tonalá Jalisco, México; beatriz.castillo@udg.mx; Tel.:+527772708001 9 2 Université de La Rochelle, La Rochelle. Laboratoire des Sciences de l’Ingénieur pour l’Environnement, 10 Avenue Michel Crépeau 17042 La Rochelle Cedex 1 – France; kallaf@univ-lr.fr 11 3 Coordinación de Refrigeración y bombas de calor y sistemas energéticos, Instituto de Energías Renovables, 12 Universidad Nacional Autónoma de México. Privada Xochicalco s/n, Colonia Centro, 62580, Temixco, 13 Morelos, México; ipf@ier.unam.mx, rbb@ier.unam.mx, wrgf@ier.unam.mx. 14 2 Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp), Universidad Autónoma del Estado 15 de Morelos, Av. Universidad No. 1001, Col Chamilpa, Cuernavaca, Morelos, México. C.P. 62209; 16 rosenberg@uaem.mx 17


Introduction
The desalination process aims to obtain water suitable for human consumption (Castillo and Pilatowsky, 2013), which is important but cost-and energy intensive process (Erlbeck et al., 2017).
Numerous methods have been used, nevertheless, until June 2016, the 51% of seawater on the world, was desalted by a Multi-Stage Flash (MSF), a thermal process accompanied with high environmental Freezing/Melting (F/M) has the advantage to be the most theoretically energy-efficient desalination processes.This low consumption is due to the fact that the latent heat of melting of the ice is only 334 kJ/kg against 2,257 kJ/kg for its vaporization, so the process of the F/M could save up to 70% of the energy required by the thermal conventional desalination processes (Williams et al., 2015), (Mtombeni, T., Maree, J.P., Zvinowanda, C, 2013).The most commercial process is Reverse Osmosis, which is the more used technology for brackish water desalting, near to 84% on the world, (IDA Desalination, 2017).However, the problem of membrane replacement persists, no being the case of the F/M process.
The F/M process can remove dissolved salts in solutions during the formation of ice crystals (Johnson, 1976;Jungwirth, 2005;Blank, Tusel and Nisanc, 2007).Freezing has been used to separate a wide variety of contaminants from water, such as minerals, organic chemicals and dissolved particles.In addition to low energy consumption, it operates at low temperature, minimizing corrosion problems and allowing low cost materials such as plastic.A pre-treatment is not required, thus avoiding the use of chemicals and implies minimum environmental impact (Rahman et al. 2007, (Johnson, 1979;Mandri et al., 2011).
During the decades of the 1960s and 1970s, there were several attempts to commercialize this technology, which has continued its development for 45 years, through many technological innovations (Thompson and Nelson, 1954;Marshall, Goff and Hartel, 1967;Curran, 1970;Rice and Chau, 1997) and countless pilot plants.
However, this operation stays having many drawbacks compared to reverse osmosis and thermal processes, such as high capital costs.However, the most difficult problem to solve has been the entrapment of salts in the ice during the crystallization, hindering their separation, and requiring for it, crushing and recrystallization of the ice, which increases operating costs (Wiegandt and Berg, 1980;Subramani and Jacangelo, 2015).
Ice quality depends on several factors linked mainly to freezing process kinetics.The objective is to eject salt in a small volume of unfrozen brine.However, it often occurs that some salts are trapped in the ice crystal, even independently of the solubility of the salts in ice (Jungwirth, Vrbka and Jungwirth, 2005).Research works have been oriented towards reaching a lower water salinity.Thus, Yoshihito Shirai proposed to use agitators and fan to circulate the solution and cold air to improve heat and mass transfer, additionally an seed ice was integrated (Shirai et al., 1998).(Yu, Ma and Zhang, 2007) find that at low initial solution temperature (about 6oC), the temperature of coolant, the solution concentration and the re-crystallization process as important factors to obtain a mayor purity of the ice produced.Experiments have also been conducted at different salt concentrations, stirring velocity in the separation vessel, as well as increase the number of washing steps to determine if the mentioned parameters can reduce the effects on salt retention.
In an experimental study, the desalination rate positively correlated with freezing temperature while negatively correlated with freezing rate, the solid fraction, solution concentration and the surface area of the freezing container (Luo et al. 2010).Reddy et al proposed that an agitator can reduce the salts encapsulated in crystals (Reddy et al., 2009).(Erlbeck et al., 2017), carried out important research on the effect on efficient desalination with different methods as cooled plate with several operation conditions and their dependence on ice production, the heat transfer via droplets of non-miscible organic fluids and classical crystallization with stirred, then, they add a post treatment to produce usable water.

Theoretical studies
Some authors introduced the conditions prevailing in the interface, the latent heat and the rejection of the salts were accomplished (Körber, Scheiwe and Wollhöver, 1983).The position and advancement of the interface is not known a priori but should be determined through the mechanisms of mass transport and brine properties depending on concentration.
Other theoretical works studied the combination of thermal behavior, heat transfer and liquid-solid phase-change.(Voller and Cross, 1981), describe an extension to the explicit finite difference solution to the enthalpy formulation, which describe an accurate solution to Stefan problems.The temperature distribution, the phase of change position and its velocity, using a finite difference method for a melting problem for different materials were evaluated (Savović and Caldwell, 2003).
A complete list of studies of phase-change phenomena from solid to liquid or solid to gas was presented (Yao and Prusia, 1989).
In other research, mathematical models used finite differences to approximate the crystal growth and the temperature distribution, as well as the position and velocity of the interface in motion (Orcutt, 1969), in relation to solute distribution during freezing, (Terwilliger and Dizio, 1970;Körber, Scheiwe and Wollhöver, 1983).The solute distribution coefficients were also determined.Measurements of temperature and concentration profiles were performed (Weeks and Lofgren, 1967;Grange, Viskanta and Stevenson, 1976;Wollhöver et al., 1985;Kapembwa, Rodríguez-Pascual and Lewis, 2014).The phenomenon of solute or impurity rejection at the solid-liquid interface stayed far from understanding (Grange, Viskanta and Stevenson, 1976).However, most of the works do not consider the physical properties changes of the saline solution which have no constant concentration, leading to inaccuracies in their calculations.The present work aims to analyze profiles of both temperature and salinity applying the mains heat and mass transfer equations, providing information to promote the diffusion of the salt, by the temperature gradient.Other factors as the initial saline concentration, recipient dimensions or geometry for instance should be considered to obtain a better of salt separation from ice being the temperature gradient the most important to produce an acceptable fresh water quality.This process has not been widely commercially used for desalination purposes but however this technology had been applied in the food Industries.

Physical model
Based on the literature reviewed a lack information about saline solution freezing and salt rejection was found.Especially the effect of concentration gradient on the physical properties of the solution, such as: density, freezing point, thermal conductivity, thermal coefficients, kinematic viscosity, etc. consequently, a physical model is proposed, which allows their calculation.

Description
This physical model consists in a horizontal transparent cylinder which was thermal insulated except in one flat side to drive the heat flow in a unidimensional x-direction.The non-insulated surface was exposed to the cold air at temperatures Ta below the freezing point of saline solution.Depending on both, brine and air temperatures; Tb and Tair, respectively, ice formation evolution take place at the solid-liquid interface.When Tair is below saline solution freezing point and Tb higher than the temperature of interface surface of ice, the heat transfer should be performed from brine towards the cold air.The heat transfer is carried out by natural convection between the brine and the ice formed and between the cylinder surface and cold air.By conduction through the ice and the cylinder material thickness.Figure 1 shows the container of saline water.Ice formation should be performed progressively so that it allows to transfer the ions to the brine.

Main equations
Freezing cannot be simulated as a simple general equation of heat conduction, which only governs the solid phase.For the sensible heat transfer, the model is normally formulated for one-dimension thermal conduction.At x=0, convection heat transfer is performed between cooling air and cylinder surface no insulated, with a cooling rate: Eq.1 At time t, freezing interface is located at x=xFr and Eq.1 is only adequate for 0≤x<xFr.At x=xFr, the convection heat transfer is performed between brine and ice surface: Combining the equations 1, and 2 (Eq.3), the solution of the evolution of temperature and global heat flow as function of x and t is obtained.The progression of the position of ice-brine interface xo=xi and the heat flow allows identifying the development of ice production as function of time, at xo=xi: Where is the rate of volume of ice formed per unit area on the growing surface (m 3 /(s m 2 )) and ρ is the latent fusion heat per volume (J/m 3 ).When the distribution of temperatures allows that the total generated heat flow to be negative, ice rate increases and consequently is positive.
An adequate analytic solution of these coupled partial differential equations is very difficult and may be obtained only for specific cases.Moreover, the particular solution is known at the actual physical conditions on the border.The repartition of temperature should modify the physical and thermal properties within the ice.In order to simplify the solution and adopt practical values, it is assumed that the homogeneity of physical properties of the subcooled solid phase (ice), such as density ρice, heat capacity cice, conductivity kice, and thermal diffusivity i).Further simplifications were implemented by assuming that the cooling air temperature Tair keeps constant, while liquid (brine) temperature Tb kept slightly higher than solidification temperature of water (To=TFr≈0 °C) and lower than brine freezing.
The temperature gradients ( − T ) = ( − T ) generated, the heat flow rate; acting as thermal resistances in series from the ice, the plate, and the air: The heat flow rate from the brine section is given by Eq. 2, it is exclusive from convection heat transfer.
Combining the equations 2 and 4 (in Eq. 5) to vanish the heat flow rates: It manages to relate the ice depth xo to the freezing time to.If the process starts at t=0 and continue until tfr time, dtfr needs to increase dx of ice generation becomes: This Eq. 6 is used to identify the time tfr necessary to freeze a thickness xo of brine.Obviously, the increase of ice thickness as function of time implies that the coefficient of dxo should be positive.

Complementary equations
In order to obtain the thermal properties, (Yunus Cengel, 2011), it was used the following relations: Eq.7 It is a function of: Eq.9 And then, finally: For both air and brine sides, respectively.
While saline solution freezing process occurs the salt contained homogeneously at the beginning of the process is displaced towards the non-frozen section, consequently, the concentration increases as ice (now free of salt) grows, and the factors affecting the heat transfer phenomena (described in Eqs. 7-11), depend on the characteristics of the actual solution.To simplify the model and give an accurate approach, we used sodium chloride (NaCl) to estimate the brine behavior as the evolution of seawater, sodium chloride is the main salt in seawater and values for density, dynamic viscosity and specific heat are in good agreement with literature values of sea water (Melinder and Ignatowicz, 2015).The freezing temperature (° ) of the brine salt solution was estimated using the equation proposed in (Fofonoff and R.C., 1983): The characteristics of dilute solutions, depend on salt concentration.Thus, the freezing point decreases, the density increase, such as the coefficient of volumetric expansion and osmotic pressure.
Given the lack information and because the literature gives no values for the variables to be used below 0 o C, the functions have been extrapolated mainly based on the freezing point and the properties of sodium chloride-water solutions.The values for the modified properties during the freezing process, were obtained from the literature, (Å.Melinder, 2007;Cengel and Boles, 2011;Frank Dreith, Raj M. Manglik, 2011), adapting for low temperatures for air and NaCl-water solution, shown in Figure 2  The values of kinematic viscosity, thermal conductivity which depend on brine concentration during the process are defined by many authors.(Å.Melinder, 2007;Sharqawy, Mostafa H., Lienhard V, John H., 2013;Melinder and Ignatowicz, 2015).Their results agree with measurements of thermal conductivity reported by Ryedel, (Granryd and Melinder, 2005;A. Melinder, 2007)  Prandtl number (Pr) for air low temperature, kinematic viscosity and thermal conductivity can be obtained by many ways.In this work the tables presented by Holman (1998) are used.

Salt diffusion analysis
For salt diffusion during freezing, the model proposed by Allaf is used (Allaf et al., 2011), issued from Fick law, (Allaf, 1982): The salinity of solution is , , subsequently, if it is assumed that the water velocity in the brine was globally: v ≈ 0, and the movement of salt should be uniaxial towards one dimension; Eq.18 became: Since the grow ice aims reaching a new equilibrium phase change (Fukui and Maeda, 2002), solidification process is responsible of a partial or total expulsion of salt towards the residual more concentrated brine.The requirement of a complete expulsion of salt ions from dx layer of ice implies that ( , ) should be twice that ( , ) , then: Eq.22 and: = − Eq.23 Thus, the time dtfr needed for diffusing salt out of dx layer should be: Accordingly, the study of mass transfer requires estimation of diffusivity D of salt in the brine.There are several accurate experimental methods for the measurement of D, for example, optical (Krahn, Schwelger and Lucas, 1983), spectroscopic (Stejskal and Tanner, 1965) the Taylor method (Ouano, 1972;Atwood and Goldstein, 1984).However, the value of D has frequently been estimated because of the problems due the expensive instrumentation.

Eq.25
The limiting ionic conductance in water at 298 K for Cl -is 76.3 and for Na + 50.1 (A/cm 2 )V/cm)(gequiv/cm 3 ) (Harned and Benton B., 1943).The Table 2 presents some result of these interesting parameters.Excessive gradient of temperatures will result on entrapment of salts on ice and contrary, the diffusion occurs at low freezing velocity.S: Salt concentration on liquid.

Conclusions
New fundamental model was defined to obtain the conditions able to achieve a suitable saline solution desalination.This model is based on the comparison of both freezing process and salt diffusion in saline water.It involves the effect of thermal parameters, which normally depend on salt concentration (conductivity, freezing temperature, etc.), physical characteristics (density, expansion coefficient, kinematic viscosity, salt diffusivity, etc.), and equipment geometry.The model is at the same time simple and effective and provides an approach tool to conduct F/M desalination process.
Important information has been collected to achieve data closer to observed in experimentation in a complex problem.The ice grow rate value must be around 3E-07 to ice salt separation and the freezing temperature must be lower than -15 o C to achieve 233 g/l on liquid.The results of this model will help the design and build high-performant adequate F/M prototypes and industrial plants.

Figure 1 .
Figure 1.Scheme of the physical model

Figure 2
Figure 2 Kinematic viscosity for NaCl-water solutions Figure 3 Air kinematic viscosity at low temperatures NaCl-water solution, respectively.

Figure 4
Figure 4 Thermal conductivity for NaCl-Water solutions at different salinity

Figure 7
Figure 6 Prandtl Number for low temperatures Figure 7 Salinity and freezing temperature evolution