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

Abstract

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 efficacy 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 coefficient. 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.

1. 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 nanoparticles. Also, to predict the thermal conductivity of mixture of liquid film and nanoparticles, Xuan and Roetzel [8] suggested two different protocols. Ali et al. [9] investigated the transfer of heat and mass through a falling solution. They found controversial results, which suggest having a big base of data.

Several other works studied the hydrodynamic and thermal behavior of nano fluid numerically and experimentally in confined flows. Pak and Cho [10] observed the transfer of heat by forced convection when a tube is heated for several Reynolds numbers. They prove that the coefficient of convective heat transfer enhances with the volume rate of nanoparticles. To evaluate the Nusselt number when the regime is turbulent, they provide an approximate correlation. Experimental investigation of the heat transfer and the characteristics of flow in a tube, heated externally with a constant flux, is performed by Li and Xuan [11]. The collected data of the mixtures: (water and Cu, TiO₂ particles) and (water and Al₂O₃ particles), provided an empirical correlation for the Nusselt number. Theses approximations were given when the flows are laminar and turbulent. Under laminar flow regime, the intensity of the heat transfer through a water-Al₂O₃ mixture is studied by Wen and Ding [12]. For copper tube, they found that an augmentation of Reynolds number and the nano fluid volume concentration ameliorate the heat transfer coefficient by convection. Yang et al. [13] experimentally evaluated the transfer of heat by the convection of nano fluids formed of transmission fluids and graphitic-based nanoparticles. It is shown that the presence of nanoparticles engenders a real augmentation of the heat transfer through the physical systems. Fohanno. et al. [14] presented a critical investigation of the different numerical methods to compute the thermophysical properties of several nano fluids [15,16,17,18,19].

By using a pendant droplet method, Chen et al. [20] evaluated the tension of surface relative to a three types of nano fluid (laponite-water, silver-water and Fe₂O₃-water). Orejon et al. [21] studied nano fluid droplets evaporating under DC conditions. They revealed several evolutions relative to contact angle, deposit patterns and contact radius. It is shown that if we apply the DC potential, the TiO₂ nano fluids receding contact is smooth and continuous. Also, the deposit patterning is uniform. Furthermore, the nanoparticles interactions are different compared to the DC potential case. Ni et al. [22] tested the evaporation of nano fluid based on solar energy. They show that the performance of this system is around 69% relative to the nano fluid evaporation and the transient phase change depends on the nanoparticles surface chemistry. In addition, they use the Fourier law to model the thermal attitude of the nano fluid at microscale. In the same field, Amjad et al. [23] used gold nanoparticles to generate steam under solar heating. They demonstrate that a cooled bulk fluid causes an important surface of evaporation and they evaluate the photothermal performance by integrating a new method. They demonstrate, experimentally, that a 0.04% of gold in volume increases the evaporation rate by 95%. Zhou et al. [24] analyzed the absorption and desorption of molecules of fluid in a porous media formed by nanoparticles. Their numerical results are consistent with experimental data and prove that we can modify the properties of the energy storage by exothermic and endothermic adsorption and desorption. Moshizi [25] studied the influence of a constant magnetic field in the heat and mass transfer through a micro channel in presence of cu-water nano fluid. The Brinkman model allowed him to investigate the problem and to transform the nonlinear equations to the simple linear system. Moshizi et al. [26] gave a theoretical study of the convective transfer of heat taking into account the pressure drop. They consider the effect of the nanoparticle volume rate by using the Buongiorno model. S. A. Moshizi [27] studied the influence of the (MHD) field on the developed forced convection of heat with nano fluid in pipes. S. A. Moshizi and A. Malvandi [28] looked mathematically for the effect of an external magnetic field on the nanoparticles motion. Their study was during a mixed gas convection inside micro-channels. They consider the effect of the nanoparticle volume rate by using the Buongiorno model. The same authors [29] related the effect of the nano particles migration on the heat exchangers efficiency. With changing the migration modes of the Al₂O₃ inside water and under the mixed convection of gas in vertical micro channels, they demonstrate the influence of the thermophysical variations on the heat transfer. S. A. Moshizi et al. [30] present a study of magnetic field direction on transfers. They vary this direction in the case of the (MHD) nano fluids in isothermal micro tubes. This (MHD) phenomena was 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).

2. 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].

• The nano film enters the channel with a temperature T_nf0, a thickness δ₀ and a velocity u_nf0. The air enters with a temperature T₀, mass fraction vapor c₀ and a velocity u₀. Assumptions are used to build up the mathematical system of the problem.
• The flows through the vertical channel are laminar and stationary (gas and film). The study of the transfers is in two-dimensional case.
• The nano film is considered as a boundary layer.
• Humid air is considered as an ideal mixture and perfect gas.
• We don’t take into the account the surface tension.
• The gas–nano fluid interface is in the thermodynamic equilibrium.
• We neglect the Soret and Duffour effects.
• 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.

2.1. 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:

Y = (y − δ)/(d − δ); X = x/H

(1)

In liquid phase:

Y_nf = (y − δ)/δ; X = x/H

(2)

Adopting these transformations, the equations governing the flow and the heat and mass transfers in the liquid and gas phases are:

2.1.1. For Liquid Phase

Continuity equation

$$\frac{\partial {\rho }_{nf}{u}_{nf}}{\partial X}-\frac{Y_{nf}}{\delta }\frac{\partial \delta }{\partial X}\frac{\partial {\rho }_{nf}{u}_{nf}}{\partial Y_{nf}}+\frac{H}{\delta }\frac{\partial {\rho }_{nf}{v}_{nf}}{\partial Y_{nf}}=0.$$

(3)

Momentum equation

$$u_{nf}\frac{\partial u_{nf}}{\partial X}+\left(u_{nf}\frac{Y_{nf}}{\delta }\frac{\partial \delta }{\partial X}+v_{nf}\frac{H}{\delta }\right)\frac{\partial u_{nf}}{\partial Y_{nf}}=\frac{H}{{\rho }_{nf}{\delta }^2}\frac{\partial }{\partial Y_{nf}}\left({\mu }_{nf}\frac{\partial u_{nf}}{\partial Y_{nf}}\right)+gH$$

(4)

Energy equation

$$u_{nf}\frac{\partial T_{nf}}{\partial X}\left(u_{nf}\frac{Y_{nf}}{\delta }\frac{\partial \delta }{\partial X}+v_{nf}\frac{H}{\delta }\right)\frac{\partial T_{nf}}{\partial Y_{nf}}=\frac{H}{{\rho }_{nf}{C}_{pnf}}\frac{1}{{\delta }^2}\frac{\partial }{\partial Y_{nf}}\left({\lambda }_{nf}\frac{\partial T_{nf}}{\partial Y_{nf}}\right)$$

(5)

Conservation of liquid mass flow rate:

$${{\displaystyle \int }}_0^1\delta {\rho }_{nf}{u}_{nf}d{Y}_{nf}=\left[m_{nf}-H{{\displaystyle \int }}_0^X\rho v\left(X,Y=1\right)\right]dX$$

(6)

2.1.2. For Gas Phase

Continuity equation

$$\frac{\partial \rho u}{\partial X}+\frac{Y-1}{d-\delta }\frac{\partial \delta }{\partial X}\frac{\partial \rho u}{\partial Y}+\frac{H}{d-\delta }\frac{\partial \rho u}{\partial Y}=0$$

(7)

Momentum equation

$$\begin{array}{c}u\frac{\partial u}{\partial X}+\left(u\frac{Y-1}{d-\delta }\frac{\partial \delta }{\partial X}+v\frac{H}{d-\delta }\right)\frac{\partial u}{\partial Y}\hfill \\ =-\frac{1}{\rho }\frac{dp}{dX}-\frac{1}{\rho }\frac{Y-1}{d-\delta }\frac{\partial \delta }{\partial X}\frac{\partial p}{\partial Y}-\beta gH\left(T-T_0\right)\hfill \\ -{\beta }^{*}gH\left(c-c_0\right)+\frac{1}{\rho }\frac{H}{{\left(d-\delta \right)}^2}\frac{\partial }{\partial Y}\left(\mu \frac{\partial u}{\partial Y}\right)\hfill \end{array}$$

(8)

Energy equation

$$u\frac{\partial T}{\partial X}+\left(u\frac{Y-1}{d-\delta }\frac{\partial \delta }{\partial X}+v\frac{H}{d-\delta }\right)\frac{\partial T}{\partial Y}=\frac{1}{\rho C_p}\left(\frac{H}{{\left(d-\delta \right)}^2}\right)\frac{\partial }{\partial Y}\left(\lambda \frac{\partial T}{\partial Y}\right)+\rho D\left(c_{pv}-c_{pa}\right)\frac{\partial T}{\partial Y}\frac{\partial c}{\partial Y}$$

(9)

Diffusion equation

$$u\frac{\partial c}{\partial X}+\left(u\frac{Y-1}{d-\delta }\frac{\partial \delta }{\partial X}+\frac{H}{d-\delta }v\right)\frac{\partial c}{\partial Y}=\frac{1}{\rho }\frac{1}{{\left(1-\delta \right)}^2}\left(\frac{\partial }{\partial Y}\left(\rho D\frac{\partial c}{\partial Y}\right)\right)$$

(10)

Conservation of gas mass flow rate

$${{\displaystyle \int }}_0^1\rho u\left(X,Y\right)\left(d-\delta \right)dY=\left[\left(d-{\delta }_0\right){\rho }_0{u}_0+H{{\displaystyle \int }}_0^X\rho v\left(X,Y=0\right)dX\right]$$

(11)

2.2. Boundary Conditions

For X = 0 (channel entry)

$$u=u_0;T=T_0,c=c_0;T_{nf}=T_{nf0};\mathsf{\delta }\left(0\right)={\mathsf{\delta }}_0;{\mathrm{u}}_{\mathrm{nf}}={\mathrm{u}}_{\mathrm{nf0}}$$

(12)

For Y = 1 (dry plate)

$${\left[-\lambda \frac{1}{d-\delta }\frac{\partial T}{\partial Y}\right]}_{Y=1}=h\left(T\left(X,1\right)-T_0\right) \mathrm{and}{\frac{\partial c}{\partial Y})}_{Y=1}=0$$

(13)

For Y = 0 (interface between the two phases) (in the interface y = δ then returning to the Equation (1), y = δ give Y = 0)

For the case where we neglect the normal stress forces

$$\frac{1}{d-\delta }\mu {\left(\frac{\partial u}{\partial Y}\right)}_{Y=0}=\frac{1}{\delta }{\mu }_{nf}{\left(\frac{\partial u}{\partial Y_{nf}}\right)}_{Y_{nf}=1}$$

(14)

The equality of the velocity for the two phases

$$u\left(X,Y=0\right)=u_{nf}\left(X,Y_{nf}=1\right)$$

(15)

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), y = δ give Y = −1)

$$v(X,Y=0)={{\displaystyle \left(\frac{D}{1-c(X,0)}\frac{1}{\left(d-\delta \right)}\frac{\partial c}{\partial Y}\right)}}_{\left(Y=0\right)}$$

(16)

$$T\left(X,Y=0\right)=T_{nf}\left(X,Y_{nf}=1\right),c\left(X,Y=0\right)=c_{sat}(T\left(X,Y=0\right)$$

(17)

$$-\frac{1}{\delta }{\lambda }_{nf}{\left(\frac{\partial T_{sw}}{\partial Y_{nf}}\right)}_{Y_{nf}=1}=-\lambda \frac{1}{d-\delta }{\left(\frac{\partial T}{\partial Y}\right)}_{Y=0}-\frac{1}{d-\delta }\frac{\rho L_v\left(D+\overline{D}\right)}{\left(1-c\left(X,1\right)\right)}{\left(\frac{\partial c}{\partial Y}\right)}_{Y=0}$$

(18)

For Y_nf = −1 (heated plate), regarding the Equation (2)

$$u_{nf}=0,-{\lambda }_{nf}{\left(\frac{\partial T_{nf}}{\partial Y}\right)}_{Ynf=0}=q_w$$

(19)

where $c_{sat}\left(T\left(x,Y=0\right)\right)=\frac{M_v{P}_{vs}}{M_v{P}_{vs}-M_a\left(p-p_{vs}\right)}$ 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:

$$m\left(X\right)=\frac{\rho D}{1-c\left(X,0\right)}\frac{1}{d-\delta }{\left(\frac{\partial c}{\partial Y}\right)}_{Y=0}$$

(20)

$$m_t\left(X\right)={{\displaystyle \int }}_0^{X}m\left(X\right)dX$$

(21)

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.

2.3. 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]:

• λ_nf = λ _water(1.72x_v + 1)
• µ_nf = µ _water(123x_v² + 7.3x_v + 1)

For the density, specific heat, and the volume expansion coefficient, we use the usual rules of mixture [14]:

• ρ_nf = (1 − x_v) ρ_water + x_vρ_n
• (ρC_p)_nf = (1 − x_v)(ρC_p)_water + x_v(ρC_p)_n
• (ρβ)_nf = (1 − x_v)(ρβ)_water + x_v(ρβ)_n
• (ρβ*)_nf = (1 − x_v)(ρβ*)_water + x_v(ρβ*)_n

The thermophysical properties of water and the gas mixture are obtained from [19]

2.4. Validation and Mesh Stability

The validations are down under the following conditions: (q_w = 1000 w/m², T₀ = 20 °C, T_nf = 20 °C, d = 0.015 m, H = 0.2 m, u₀ = 0.5 m/s, relative humidity equal to 50% and m_0L = 0.01 kg·m⁻¹s⁻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. Stability of calculation from a mesh variations-interfacial temperature.

POSITION AXIALE	(121,121,81)	(121,121,21)	(121,51,41)	(121,51,21)	(71,51,21)	(51,41,21)
X = 0.2	333.73	333.65	333.45	333.42	333.60	333.39
X = 0.4	334.96	334.92	334.81	334.73	334.91	334.69
X = 0.6	337.43	337.39	337.29	337.22	337.12	337.36
X = 0.8	339.06	339.02	338.88	338.77	338.71	338.98
X = 1.0	340.35	340.31	340.18	340.11	340.02	340.28

, 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.

3. 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/Km². In this study, we present the effect of the nano fluid volume rate on the transfers when the film plate is heated. The simulations are done under the following conditions: (T₀ = 25 °C, T_nf = 25 °C, d = 0.015 m, H = 3 m, u_o = 1 m/s, c₀ = 0 and m_0L = 0.01 kg·m⁻¹s⁻¹).

At present, the film plate is heated by a uniform heat flux q_w = 1000 W/m². 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)).

4. 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.