Evaporation of Water / Alumina Nanoﬂuid Film by Mixed Convection Inside Heated Vertical Channel

: In industrial devices like heat recovery systems, heat pumps, as well as symmetric and complex engineering systems, a nano ﬂuid mixture is used. Regarding the nature of the energy sources (thermal or thermal and electrical), many physical systems could represent possible applications in manufactural activities. The presence of nanoparticles inside a solvent is of great interest in order to optimize the e ﬃ cacy of the nano-technology systems. The present work deals with heat and mass transfer through a vertical channel where an alumina / water ﬁlm mixture ﬂows on one of its plates. For simulation, we use a numerical method under mixed convection during water / alumina nano ﬂuid evaporation. We heat the ﬂown plate uniformly while the other is dry and exchange heat with a constant coe ﬃ cient. The gas mixture enters channel with a constant proﬁle. Results show that an augmentation of the volume rate of the nanoparticle disadvantages evaporation if the heating is absent. Otherwise, if the heating exists, an increasing volume rate of the nanoparticle advantages evaporation. We found also that the ﬁlm velocity behavior when the volume rate of the nanoparticle varies, independent of the heating.


Introduction
Because of its small thickness, the evaporation of thin film provides high heat transfer rates. This phenomenon appears in micro heat pumps, MEMS systems (Microelectromechanical Systems), and in modern applications of nanotechnology devices (Wang et al. [1]; Plawsky et al. [2]). As an efficient cooling liquid, nano fluid enhances the transport of heat during thin film evaporation (Do and Jang [3]). Micro channel configuration study of nano fluid presence effects on the transfers was performed by Jun-Jie Zhao et al. [4]. It was shown that the nano fluid thermophysical property effect is mainly due to thermal conductivity enhancement. They conclude that increasing Cu-volume fraction engenders that condensation augments and consequently Cu-solutions becomes more stable.
Other works concerned the investigation of the thermophysical properties of the nano fluids. Choi [5] showed the amelioration of the conductivity by introducing nanoparticles. A protocol of measuring the thermal conductivity of a nano fluids was performed by Lee et al. [6]. By using a hot-wire apparatus, Xuan and Li [7] evaluated the thermal conductivity of mixture of liquid film and investigated when the nano fluid is electrically conductive in micro annuli [31]. M. Sohail et al. [32] analysis the entropy relative a Maxwell nano fluid by modifying the heat and mass transfer models. Also, the generalize the law of Fick. I. M. Eldesoky et al. [33] showed that position and time affect the magnetic field influence on the temperature of nano fluid. S. I. Abdelsalam et al. [34] studied the unsteady peristaltic motion of a non-Newtonian nano fluid. They show the influence of a magnetic field and the currents of Hall. In addition, they take into the account the chemical reaction.
Regarding the previous studies, no study has given interest to the behavior of nano fluids in a semi-confined area like vertical channel. It will be the objective of this present work when no external heat is provided (heated flown plate case).

Analysis
This work provides a simulation study of heat and mass transfer when evaporation by mixed convection in a vertical channel (Figure 1) formed by two parallel plates. We heat the flown one with a constant flux. An alumina-water nano fluid is falling down in the presence of descending laminar mixed convection flow. The second plate is dry and exchanges heat with environment. The thermophysical properties of the nano fluid are obtained from [14] and those of the humid air from [15].

I.
The nano film enters the channel with a temperature T nf0 , a thickness δ 0 and a velocity u nf0 . The air enters with a temperature T 0 , mass fraction vapor c 0 and a velocity u 0 . Assumptions are used to build up the mathematical system of the problem. II.
The flows through the vertical channel are laminar and stationary (gas and film). The study of the transfers is in two-dimensional case. III.
The nano film is considered as a boundary layer. IV.
Humid air is considered as an ideal mixture and perfect gas. V.
We don't take into the account the surface tension. VI.
The gas-nano fluid interface is in the thermodynamic equilibrium. VII. We neglect the Soret and Duffour effects. VIII. In the energy equation, we don't consider the pressure work terms, Radiation heat transfer and the viscous dissipation.
Firstly, we will present the general equations of conservation. Then, the boundary conditions equations will be presented in detail.

General Equations
The heat and mass transfer for the laminar mixed convection induced by the combined thermal and mass buoyancy forces, introducing the following transformations: In gas phase: In liquid phase: Adopting these transformations, the equations governing the flow and the heat and mass transfers in the liquid and gas phases are: Continuity equation Momentum equation Energy equation Conservation of liquid mass flow rate: Momentum equation Diffusion equation Conservation of gas mass flow rate

Boundary Conditions
For X = 0 (channel entry) For Y = 1 (dry plate) For Y = 0 (interface between the two phases) (in the interface y = δ then returning to the Equation For the case where we neglect the normal stress forces The equality of the velocity for the two phases We consider that the film thickness variation is negligible, and the rate of phase change is small: (interface between the two phases) (in the interface y = 0 then returning to the Equation (1) For Y nf = −1 (heated plate), regarding the Equation (2) where is a vapor concentration at saturation. P vs is a vapor pressure at saturation (Vachon [16]).
In addition, we define the local evaporation rate m(X) and the total evaporation rate at the interface m t at the axial location X as: The global mass balance at each section and the boundary conditions closes the mathematical system. The present problem defined by the system of equations with the boundary conditions is Appl. Sci. 2020, 10, 2380 6 of 11 solved numerically using a finite difference marching procedure in the downstream direction. A fully implicit scheme where the axial convection terms are approximated by the upstream difference, and the transverse convection and diffusion terms by the central difference is employed. The simulations are to describe the effect of alumina volume rate on the transfers. Particular attention is given to show the efficiency of the alumina content on the phase change temperature.

Thermophysical properties
All the thermophysical properties of the nano film are obtained from [14,18]. For the thermal conductivity and the dynamic viscosity, we use the empirical correlations of Mintsa et al. [18]: For the density, specific heat, and the volume expansion coefficient, we use the usual rules of mixture [14]: The thermophysical properties of water and the gas mixture are obtained from [19]

Validation and Mesh Stability
The validations are down under the following conditions: (q w = 1000 w/m 2 , T 0 = 20 • C, T nf = 20 • C, d = 0.015 m, H = 0.2 m, u 0 = 0.5 m/s, relative humidity equal to 50% and m 0L = 0.01 kg·m −1 s −1 ). A comparison is made with Yan et al. [17]. Figure 2 shows that the relative difference between our results and those obtained by Yan [17] is lower than 8% for the interfacial temperature. These discrepancies cold be explained by the difference between the physical properties used in the two works and the effect of convection terms neglected by Yan [17] in the liquid phase. For the mesh stability as mention in Table 1, a maximum variation observed is less than 1% for the interfacial temperature. To reduce the cost of computation, we choose the grid 71 × 5l × 21 in this study.
The global mass balance at each section and the boundary conditions closes the mathematical system. The present problem defined by the system of equations with the boundary conditions is solved numerically using a finite difference marching procedure in the downstream direction. A fully implicit scheme where the axial convection terms are approximated by the upstream difference, and the transverse convection and diffusion terms by the central difference is employed. The simulations are to describe the effect of alumina volume rate on the transfers. Particular attention is given to show the efficiency of the alumina content on the phase change temperature.

Thermophysical Properties
All the thermophysical properties of the nano film are obtained from [14,18]. For the thermal conductivity and the dynamic viscosity, we use the empirical correlations of Mintsa et al. [18]: • λnf = λ water(1.72xv + 1) • µnf = µ water(123xv 2 + 7.3xv + 1) For the density, specific heat, and the volume expansion coefficient, we use the usual rules of mixture [14]: The thermophysical properties of water and the gas mixture are obtained from [19]

Validation and Mesh Stability
The validations are down under the following conditions: (qw = 1000 w/m 2 , T0 = 20 °C, Tnf = 20 °C, d = 0.015 m, H = 0.2 m, u0 = 0.5 m/s, relative humidity equal to 50% and m0L = 0.01 kg.m −1 s −1 ). A comparison is made with Yan et al. [17]. Figure 2 shows that the relative difference between our results and those obtained by Yan [17] is lower than 8% for the interfacial temperature. These discrepancies cold be explained by the difference between the physical properties used in the two works and the effect of convection terms neglected by Yan [17] in the liquid phase. For the mesh stability as mention in Table 1, a maximum variation observed is less than 1% for the interfacial temperature. To reduce the cost of computation, we choose the grid 71 × 5l × 21 in this study.

Results and Discussions
We study the evaporation of liquid film by the mixed convection of a humid air. This liquid film is falling down on one of two vertical channel plates, which is heated. The second plate is dry and exchanges heat with the outlet with a certain heat transfer coefficient h = 15w/Km 2 . In this study, we present the effect of the nano fluid volume rate on the transfers when the film plate is heated. At present, the film plate is heated by a uniform heat flux q w = 1000 W/m 2 . Figure 3 shows the effect of the volume rate of the nanoparticle on the channel temperature at the exit. Because of the heating, the effect of the thermal conduction is significant and dominate the effect of density and the specific heat. At seen in Equations (9) and (17), an augmentation of the thermal conductivity induces an increase of the film temperature. Also, it augments interfacial temperature ( Figure 4) and consequently the vapor temperature. This augmentation of k exists because of the increasing of x v [14]. The saturation vapor concentration increases when we elevate the nanoparticle volume rate (Vachon [16]). This is shown in Figure 5. Figure illustrates the effect of an increasing of the nanoparticle volume rate on the evaporation production. As a consequence of interfacial vapor concentration, the gradient of concentration near the interface is elevated. This increases naturally the evaporation rate ( Figure 6). The augmentation of the film density, due to the x v increasing [14], induces a deceleration of the film (Equation (8)) as shown in Figure 7. The thermal conductivity doesn't affect the convection of the film (see the equation of momentum of the nano film-Equation (8)).

Results and Discussions
We study the evaporation of liquid film by the mixed convection of a humid air. This liquid film is falling down on one of two vertical channel plates, which is heated. The second plate is dry and exchanges heat with the outlet with a certain heat transfer coefficient h = 15w/Km 2 . In this study, we present the effect of the nano fluid volume rate on the transfers when the film plate is heated. At present, the film plate is heated by a uniform heat flux qw = 1000 W/m 2 . Figure 3 shows the effect of the volume rate of the nanoparticle on the channel temperature at the exit. Because of the heating, the effect of the thermal conduction is significant and dominate the effect of density and the specific heat. At seen in Equations (9) and (17), an augmentation of the thermal conductivity induces an increase of the film temperature. Also, it augments interfacial temperature ( Figure 4) and consequently the vapor temperature. This augmentation of k exists because of the increasing of xv [14]. The saturation vapor concentration increases when we elevate the nanoparticle volume rate (Vachon [16]). This is shown in Figure 5. Figure illustrates the effect of an increasing of the nanoparticle volume rate on the evaporation production. As a consequence of interfacial vapor concentration, the gradient of concentration near the interface is elevated. This increases naturally the evaporation rate ( Figure 6). The augmentation of the film density, due to the xv increasing [14], induces a deceleration of the film (Equation (8)) as shown in Figure 7. The thermal conductivity doesn't affect the convection of the film (see the equation of momentum of the nano film-Equation (8)).

Conclusion
We performed a study of a Nano film of alumina-water flowing on a heated plate of a vertical channel. Validation and stability study of the mesh is given after mathematical analysis. Also, we have paid attention to prove the important influence of heating on the volume rate of the nanoparticle. Simulations provide the following conclusions: • The effect of the thermal conduction is significant and dominates the effect of density and the specific heat.
• An augmentation of the thermal conductivity induces an increase of the film temperature and the interfacial temperature, and consequently the vapor temperature.
• Augmentation of k exists because of the increase of xv.
• The saturation vapor concentration is elevated when we elevate the nanoparticle volume rate.
• Augmentation of k naturally increases the evaporation rate.
• The augmentation of the film density, due to the increase of xv, induces a deceleration of the film. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.

Conclusions
We performed a study of a Nano film of alumina-water flowing on a heated plate of a vertical channel. Validation and stability study of the mesh is given after mathematical analysis. Also, we have paid attention to prove the important influence of heating on the volume rate of the nanoparticle. Simulations provide the following conclusions: • The effect of the thermal conduction is significant and dominates the effect of density and the specific heat.

•
An augmentation of the thermal conductivity induces an increase of the film temperature and the interfacial temperature, and consequently the vapor temperature.

•
Augmentation of k exists because of the increase of x v .

•
The saturation vapor concentration is elevated when we elevate the nanoparticle volume rate.

•
Augmentation of k naturally increases the evaporation rate.

•
The augmentation of the film density, due to the increase of x v , induces a deceleration of the film. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.