Liquid Nanofilms’ Condensation Inside a Heat Exchanger by Mixed Convection

Abstract

Liquid nanofilm is used in industrial applications, such as heat exchangers, water desalination systems, heat pumps, distillation systems, cooling systems, and complex engineering systems. The present work focuses on the numerical investigation of the condensation of falling liquid film containing different types of nanoparticles with a low-volume fraction. The nanofluid film falls inside a heat exchanger by mixed convection. The heat exchanger is composed of two parallel vertical plates. One of the plates is wetted and heated, while the other plate is isothermal and dry. The effect of the dispersion of the Cu or Al nanoparticles in the liquid on the heat exchange, mass exchange, and condensation process was analysed. The results showed that the heat transfer was enhanced by the dispersion of the nanoparticles in the water. The copper–water nanofluid presented the highest efficiency compared to the aluminium–water nanofluid and to the basic fluid (pure water) in terms of the heat and mass exchange.

1. Introduction

Film condensation is extensively encountered in engineering areas, such as air conditioning, drying technology, and desalination.

A study of the convective heat transfer of water containing CuO nanoparticles was presented by Asirvatham et al. [1]. They showed that the convective heat transfer coefficient of the nanofluid was enhanced with a lower volume concentration of CuO nanoparticles. Sefiane and Bennacer [2] experimentally analysed the effect of nanoparticles’ concentration on nanofluid viscosity. They showed that the dispersion of nanoparticles caused an increase in the nanofluid viscosity and disadvantaged the drop evaporation. Sheremet et al. [3] numerically studied the free fluid flow of the convective heat transfer inside a porous wavy cavity in the presence of nanofluid. They showed that the nanofluid flow and heat transfer rate were influenced by the local heat source. Gorjaei et al. [4] analysed the impact of the dispersion of Al₂O₃ nanoparticles on the heat transfer inside a three-dimensional annulus. Namburu et al. [5] presented a numerical study of the steady heat exchange and the forced flow of different nanofluids inside a circular tube. They showed that an increase in the volume concentration of the nanofluids caused a reduction in the pressure loss. Chen et al. [6] presented the effects of nanoparticles on droplet evaporation. They showed an enhancement of nanofluid evaporation in the case of the packing of nanoparticles in fluid. They showed that these nanoparticles (Laponite, Fe₂O₃, and Ag)–water droplets evaporated at different rates than the base fluid (water). The effects of the Brownian motion of nanoparticles and thermophoresis on the heat transfer during liquid film boiling were analysed by Malvandi et al. [7]. Orejon et al. [8] treated the evaporation of water droplets containing TiO₂ nanoparticles. Siddiqa et al. [9] investigated the heat exchange of nanofluid flow along a vertical surface by natural convection. It was shown that the dispersion of the nanoparticles ameliorated the heat exchange. Orejon et al. [10] presented an analysis of water droplets’ evaporation containing nanoparticles of titanium dioxide (TiO₂) under direct current conditions. They showed that the TiO₂ nanofluids’ receding contact was continuous and smooth when direct current potential was applied. Perrin et al. [11] analysed liquid drop evaporation containing a low concentration of nanoparticles. A comparison between the experimental and theoretical results of the evaporation of nanofluid liquid drops was presented. Huang et al. [12] experimentally analysed liquid film evaporation. They showed that the evaporation was enhanced with an increase in the inlet temperature and inlet air flow rate. Yan [13] numerically investigated falling liquid film evaporation. Askounis et al. [14] studied water droplets’ evaporation containing a low concentration of Al₂O₃ nanoparticles. They concluded that the dispersion of Al₂O₃ nanoparticles to the water droplets with a low concentration did not affect the rate of the evaporation of droplets. Sohail et al. [15] presented a finite element analysis for ternary hybrid nanoparticles on thermal enhancement in pseudo-plastic liquid through a porous stretching sheet. They show that the thermal performance was enhanced by the mixing of nanoparticles in the base fluid. Nazir et al. [16] presented a finite element analysis for thermal enhancement in a power law hybrid nanofluid. Nazir et al. [17] studied the thermal and mass species transportation in tri-hybridised Sisko martial with a heat source over a vertical heated cylinder. Nasr et al. [18] presented a numerical study of liquid nanofilms’ evaporation inside a heat exchanger by mixed convection.

This literature review shows that a large number of studies have examined the pure liquid film condensation inside a heat exchanger. To the best of the authors’ knowledge, there has been no numerical study dedicated to analysing the liquid nanofilms’ condensation inside a heat exchanger by mixed convection. The present paper presents a numerical investigation of the heat and mass transfer during liquid nanofilms’ condensation inside a plate heat exchanger. The aim of this study is to reveal the effects of the dispersion of Cu or Al nanoparticles on the mass, heat exchange, and condensation performance.

2. Numerical Modelling

This work focused on a numerical investigation of the condensation of a liquid containing (nanofilm) different types of nanoparticles with a low volume fraction, falling inside a heat exchanger by mixed convection (Figure 1). The heat exchanger was composed of two vertical plates. The left plate was wetted and heated, while the other plate was isothermal and dry. The nanofluid film entered the plate with a mass flow rate of$m_{0L}$, a temperature of$T_{0L}$, and an inlet volume fraction of nanoparticles, ${\mathsf{\Phi }}_0$. The ambient parameters at the inlet channel were the mass fraction of water vapour, $c_0$, the velocity, $u_0$, the temperature,$T_0$, and the pressure,$p_0$.

2.1. Basic Equations

The governing equations of the nanofluid film condensation in the two phases are [5,10,12,14] are:

2.1.1. For the Liquid Phase

Continuity

$$\frac{\partial {\rho }_{nf}{u}_{nf}}{\partial x_L}+\frac{\partial {\rho }_{nf}{v}_{nf}}{\partial y_L}=0$$

(1)

x-momentum

$${\rho }_{nf}\left(u_{nf}\frac{\partial u_{nf}}{\partial x_L}+v_{nf}\frac{\partial u_{nf}}{\partial y_L}\right)={\rho }_{nf}g-\frac{d{p}_{nf}}{d{x}_L}+\frac{\partial }{\partial y_L}\left({\mu }_{nf}\frac{\partial u_{nf}}{\partial y_L}\right)$$

(2)

Energy

$${\rho }_{nf}{c}_{pnf}\left(u_{nf}\frac{\partial T_{nf}}{\partial x}+v_{nf}\frac{\partial T_{nf}}{\partial y}\right)=\frac{\partial }{\partial y}\left({\lambda }_{nf}\frac{\partial T_{nf}}{\partial y}\right)+{\rho }_p{c}_{pp}\frac{\partial }{\partial y}\left(D_B\frac{\partial \Phi }{\partial y}\frac{\partial T_{nf}}{\partial y}+\frac{D_T}{T_{nf}}{\left(\frac{\partial T_{nf}}{\partial y}\right)}^2\right)$$

(3)

Nanoparticles’ concentration

$$\frac{\partial u_{nf}\Phi }{\partial x}+\frac{\partial v_{nf}\Phi }{\partial y}=\frac{\partial }{\partial y}\left(D_B\frac{\partial \Phi }{\partial y}+\frac{D_T}{T_{nf}}\frac{\partial T_{nf}}{\partial y}\right)$$

(4)

$D_B=\beta \frac{{\mu }_L}{{\rho }_L}\Phi$, $D_T=\frac{k_{b}T}{3\pi {\mu }_L{d}_p}$ and $\beta =\frac{0.26{\lambda }_L}{\left(2{\lambda }_L+{\lambda }_p\right)}$.

2.1.2. For the Gaseous Phase

Continuity

$$\frac{\partial \rho v}{\partial y}+\frac{\partial \rho u}{\partial x}=0$$

(5)

x-momentum

$$v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}=-\frac{1}{\rho }\frac{dP}{dx}-\beta g(T-T_0)-{\beta }^{*}g(c-c_0)+\frac{1}{\rho }\frac{\partial }{\partial y}\left(\mu \frac{\partial u}{\partial y}\right)$$

(6)

Energy

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{1}{\rho c_p}\left(\frac{\partial }{\partial y}\left(\lambda \frac{\partial T}{\partial y}\right)+\rho D(c_{pv}-c_{pa})\frac{\partial T}{\partial y}\frac{\partial c_{}}{\partial y}\right)$$

(7)

Species’ diffusion

$$v\frac{\partial c}{\partial y}+u\frac{\partial c}{\partial x}=\frac{1}{\rho }\frac{\partial }{\partial y}\left(\rho D\frac{\partial c}{\partial y}\right)$$

(8)

Overall mass balance

$${\displaystyle \underset{\delta }{\overset{d}{\int }}\rho udy=(d-\delta ){\rho }_0{u}_0-{\displaystyle \underset{0}{\overset{x}{\int }}\rho }}v(x,0)dx$$

(9)

2.1.3. Boundary Conditions

• For the inlet conditions (at x = 0):

  $$u(0,y)=u_0;T(0,y)=T_0;c(0,y)=c_0$$

  (10)

  $$T_L(0,y_L)=T_{0L};m(0,y_L)=m_{0L};\Phi ={\Phi }_0$$

  (11)
• For the dry plate (at y = d):

  $$v(x=d,y)=0;u(x=d,y)=0;{\frac{\partial c_i}{\partial y})}_{y=d}=0;T(x=d,y)=Tw$$

  (12)
• For the wet plate (at y_L = 0):

  $$u_{nf}(x,0)=v_{nf}(x,0)=0;q_1=-{\lambda }_{nf}{\frac{\partial T_{nf}}{\partial y_L})}_{y_L=0}$$

  (13)
• For the gas–liquid interface (at y_L = δ and y = 0):

The continuities of the temperatures and velocities are:

$$u_{nf}\left(x,y_L=\delta \right)=u_{nf}\left(x,y=0\right);T_{nf}\left(x,y_L=\delta \right)=T\left(x,y=0\right)$$

(14)

The continuities of the shear stress give:

$${\mu }_{nf}{\frac{\partial u_{nf}}{\partial y_L})}_{y_L=\delta }=\mu {\frac{\partial u}{\partial y})}_{y=0}$$

(15)

The energy equation at the interface gives:

$$-{\lambda }_{nf}{\frac{\partial T_{nf}}{\partial y_L})}_{y_L=\delta }=-\lambda {\frac{\partial T}{\partial y})}_{y=0}-\frac{\rho L_{v}D}{1-c(x,0)}{\frac{\partial c}{\partial y})}_{y=0}$$

(16)

2.2. The Dimensionless Governing Equations

The following transformations are introduced:

In the gaseous phase: $\eta$ = (y − δ((d − δ), ξ = x/H

In the liquid phase: ${\eta }_L$= y/δ, ξ = x/H

2.2.1. For the Liquid Phase

Continuity equation

$$\frac{\partial {\rho }_L^{}{u}_L^{}}{\partial \xi }-\frac{{\eta }_L}{{\delta }^{}}\frac{\partial {\delta }^{}}{\partial \xi }\frac{\partial {\rho }_L^{}{u}_L}{\partial {\eta }_L}+\frac{H}{{\delta }^{}}\frac{\partial {\rho }_L^{}{v}_L}{\partial {\eta }_L}=0$$

(17)

x-momentum equation

$$u_{nf}^{}\frac{\partial u_{nf}^{}}{\partial \xi }+(v_{nf}\frac{H}{{\delta }^{}}-u_{nf}^{}\frac{{\eta }_L}{{\delta }^{}}\frac{\partial \delta }{\partial \xi })\frac{\partial u_{nf}^{}}{\partial {\eta }_L}=-\frac{1}{{\rho }_L}\frac{dP}{d\xi }-\frac{H}{{\rho }_L^{}{\delta }^2}\frac{\partial }{\partial {\eta }_L}\left[{\mu }_L^{}\frac{\partial u_{nf}^{}}{\partial {\eta }_L}\right]+gH$$

(18)

Energy equation

$$\begin{array}{r}{u}_{nf}\frac{\partial T_{nf}}{\partial \xi }+(u_{nf}\frac{{\eta }_L-1}{\delta }\frac{\partial \delta }{\partial \xi }+\frac{H}{\delta }{v}_{nf})\frac{\partial T_{nf}}{\partial \eta }=\frac{1}{{\rho }_{nf}{C}_{nf}^{}}\{\frac{H}{{\left(\delta \right)}^2}\frac{\partial }{\partial {\eta }_L}({\lambda }_{nf}\frac{\partial T_{nf}}{\partial {\eta }_L})\\ +{\rho }_p{C}_{pp}^{}{D}_B\frac{H}{{(\delta )}^3}\frac{\partial T_{nf}}{\partial {\eta }_L}\frac{\partial {\Phi }_{}}{\partial {\eta }_L}+\frac{{\rho }_p{C}_{pp}^{}{D}_T}{T_{nf}}\frac{H}{{(\delta )}^3}{\left(\frac{\partial T_{nf}}{\partial {\eta }_L}\right)}^2\}\end{array}$$

(19)

Nanoparticles’ concentration equation

$$\begin{array}{r}{u}_{nf}\frac{\partial \Phi }{\partial \xi }+(u_{nf}\frac{{\eta }_L-1}{\delta }\frac{\partial \delta }{\partial \xi }+\frac{H}{\delta }{v}_{nf})\frac{\partial \Phi }{\partial \eta }=\frac{1}{{\rho }_{nf}}\frac{H}{{\left(\delta \right)}^2}(\frac{\partial }{\partial {\eta }_L}({\rho }_{nf}{D}_B\frac{\partial \Phi }{\partial {\eta }_L})\\ +\frac{\partial }{\partial {\eta }_L}(\frac{{\rho }_{nf}{D}_T}{T_{nf}}\frac{\partial T_{nf}}{\partial {\eta }_L}))\\ \end{array}$$

(20)

2.2.2. For the Gaseous Phase

Continuity

$$\frac{\partial \rho u}{\partial \xi }+\frac{\eta -1}{d-\delta }\frac{\partial \delta }{\partial \xi }\frac{\partial \rho u}{\partial \eta }+\frac{H}{d-\delta }\frac{\partial \rho v}{\partial \eta }=0$$

(21)

x-momentum

$$\begin{array}{r}u\frac{\partial u}{\partial \xi }+(\frac{\eta -1}{d-\delta }\frac{\partial \delta }{\partial \xi }u+\frac{H}{d-\delta }v)\frac{\partial u}{\partial \eta }=-\frac{1}{\rho }\frac{dP}{d\xi }-g\beta H(T-T_0)-g{\beta }^{\ast }H(c-c_0)\\ +\frac{1}{\rho }\frac{H}{{\left(d-\delta \right)}^2}\frac{\partial }{\partial \eta }(\mu \frac{\partial u}{\partial \eta })\end{array}$$

(22)

Energy

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

(23)

Species’ diffusion

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

(24)

Overall mass balance

$${\displaystyle \underset{0}{\overset{1}{\int }}\rho (d-\delta )u(\xi ,\eta )d\eta }==\left[(d-{\delta }_0){\rho }_0{u}_0-H{\displaystyle \underset{0}{\overset{\xi }{\int }}\rho v(\xi ,\eta =0)d\xi }\right]$$

(25)

2.2.3. Boundary Conditions

• For the inlet conditions (at $\xi =0$):

  $$T(0,\eta )=T_0;c(0,\eta )=c_0;u(0,\eta )=u_0;p(0,\eta )=p_0$$

  (26)

  $$T_L(0,{\eta }_L)=T_{0L};{\displaystyle \underset{0}{\overset{1}{\int }}{\rho }_{0L}{\delta }_0{u}_L\left(0,{\eta }_L\right)}d{\eta }_L=m_{0L};\Phi ={\Phi }_0$$

  (27)
• For the dry plate (at η = 1):

  $$u(\xi ,1)=v(\xi ,1)=0;{\frac{\partial c_{}}{\partial \eta })}_{\eta =1}=0;T(\xi ,1)=T_w$$

  (28)
• For the wet plate (at η_L = 0):

  $$u_{nf}(\xi ,0)=v_{nf}(\xi ,0)=0;q_w=-{\lambda }_L\frac{1}{\delta }{\frac{\partial T_L}{\partial {\eta }_L})}_{{\eta }_L=0};D_B\frac{\partial \Phi }{\partial {\eta }_L}=-\frac{D_T}{T_{nf}}\frac{\partial T_{nf}}{\partial {\eta }_L}$$

  (29)
• For the gas–liquid interface (at η = 0 and η_L = 1):

  $$u_{nf}(\xi ,{\eta }_L=1)=v_{nf}(\xi ,\eta =0);T_{nf}(\xi ,{\eta }_L=1)=T_{nf}(\xi ,\eta =0)$$

  (30)

  $$v\left(\xi ,\eta =0\right)=-\frac{D}{1-c(\xi ,\eta =0)}{\frac{\partial c}{\partial \eta })}_{\eta =0},\mathrm{where}c\left(\xi ,\eta =0\right)=c_{sat}(T\left(\xi ,\eta =0\right)),$$

  (31)

  where $c_{sat}\left(T\left(\xi ,\eta =0\right)\right)=\frac{M_v}{M_g}\frac{P_{v,i}}{P_g}$.

The continuities of the shear stress give:

$$\frac{1}{\delta }{\mu }_L{\frac{\partial u_L}{\partial {\eta }_L})}_{{\eta }_L=1}=\frac{1}{d-\delta }\mu {\frac{\partial u}{\partial \eta })}_{\eta =0}$$

(32)

The energy balance at the liquid–gas interface is:

$$-\frac{1}{\delta }{\lambda }_{nf}{\frac{\partial T_{nf}}{\partial {\eta }_L})}_{{\eta }_L=1}=-\frac{1}{d-\delta }\lambda {\frac{\partial T}{\partial \eta })}_{\eta =0}-\frac{\frac{1}{d-\delta }\rho L_v{D}_{}{\frac{\partial c}{\partial \eta })}_{\eta =0}}{1-c\left(\xi ,\eta =0\right)}$$

(33)

We introduce the following quantities:

-

The latent heat flux is given by:

$$q_L=-\frac{\rho L_{v}D{\frac{\partial c}{\partial \eta })}_{\eta =0}}{\left(d-\delta \right)\left(1-c(\xi ,\eta =0)\right)}$$

(34)

-

The sensible heat flux is given by:

$$q_s=-\lambda \frac{1}{d-\delta }{\frac{\partial T}{\partial \eta })}_{\eta =0}$$

(35)

-

The local condensation rate is:

$$\stackrel{•}{m}\left(\xi \right)=-\frac{\rho D{\frac{\partial c}{\partial \eta })}_{\eta =0}}{\left(d-\delta \right)\left(1-c(\xi ,\eta =0)\right)}$$

(36)

-

The cumulated condensation rate is:

$$Mr\left(\xi \right)={\displaystyle \underset{0}{\overset{x}{\int }}\stackrel{•}{m}\left(\xi \right)d\xi }$$

(37)

The density ${\rho }_{nf}$ and heat capacity ${\left(\rho c_P\right)}_{nf}$ of the nanofluids are obtained from the following equations:

$${\rho }_{nf}=\Phi {\rho }_{{}_P}+(1-\Phi ){\rho }_{{}_L}$$

(38)

$${\left(\rho c_P\right)}_{nf}=\Phi {\rho }_P{c}_{PP}+(1-\Phi ){\rho }_L{c}_{PL}$$

(39)

The Maxwell model gives the thermal conductivity of nanofluid by:

$$\frac{{\lambda }_{nf}}{{\lambda }_L}=\frac{{\lambda }_{P+}2{\lambda }_L-2\Phi ({\lambda }_L-{\lambda }_P)}{{\lambda }_{P+}2{\lambda }_L+\Phi ({\lambda }_L-{\lambda }_P)}$$

(40)

The effective dynamic viscosity of the nanofluid is calculated according to the Brinkman model:

$${\mu }_{nf}=\frac{{\mu }_L}{{\left(1-\Phi \right)}^{2.5}}$$

(41)

3. Solution Method

The system of partial differential Equations (1)–(15) was solved using FORTRAN code with the implicit finite difference method. The governing partial differential equations were transformed into finite difference equations by using a fully implicit marching scheme. The axial convection terms were approximated by the upstream difference and the transverse convection and diffusion terms by the central difference, employed to transform the governing equations into finite difference equations. The system of finite difference equations was solved by the Gaussian elimination method. To confirm that the results were grid independent, we present in

Table 1. Stability of the calculation from the mesh variations’ evaporating rate $\stackrel{•}{m}(\xi )$ (kg.s⁻¹.m⁻²) of the water film.

I × (J + K) Grid Point	101 × (101 + 41)	101 × (71 + 41)	71 × (51 + 41)	51 × (51 + 21)	51 × (21 + 31)
ξ = 0.25	3.960	3.900	3.973	3.959	3.936
ξ = 0.50	6.651	6.530	6.572	6.593	6.567
ξ = 0.75	9.140	9.200	8.984	8.986	8.959
ξ = 1.00	11.473	11.400	11.442	11.425	11.495

the stability of the calculation from the mesh variations’ local evaporation rate $\stackrel{•}{m}(X)$. Table 1 shows that there was less than a 0.1% difference in the local evaporation rates calculated using the 71 × (51 + 21) and 51 × (41 + 21) grids. To validate the numerical simulation used in the present work, we presented in Figure 2, a comparison between our results of the interfacial (T_i − T₀) during pure water film evaporation flowing down on a vertical heated plate by mixed convection and those obtained by Yan [14]. This comparison was performed for q₂ = 0, T_0L = 298K, T₀ = 293 K, q₁ = 3000 W/m², Re = 2000, $m_{0L}=0.04kg/s$, and d = 0.015 m. Figure 2 shows a satisfactory agreement.

4. Results and Discussion

4.1. Effect of the Inlet Volume Fraction of Nanoparticles

In this work, we studied the impact of the inlet volume fraction of nanoparticles φ₀ on the mass and heat transfer during liquid film condensation flowing down on a cooled vertical plate. Figure 3 illustrates the impact of the nanoparticles’ concentration Φ₀ on the interfacial liquid–vapor temperature and vapour concentration. Figure 3a shows that the liquid–vapor interface temperature decreased with an augmentation of the inlet volume fraction of aluminium nanoparticles Φ₀. This result may be due to the high thermal conductivity and the lower heat capacity of the nanoparticles compared to the pure water (see

Table 2. Thermo-physical properties (density, heat capacity and thermal conductivity) of the aluminium and copper nanoparticles at T = 300 K [19].

Thermo-Physical Properties	Aluminium	COPPER (Cu)
ρ (kg.m−3)	2700	8933
Cp (J.kg−1.K−1)	900	385
λ (W.m−2.K−1)	240	401

). Figure 3b shows that the liquid–vapor interface concentration increased with an increase in the inlet volume fraction of aluminium nanoparticles Φ₀. This result was justified by the fact that a decrease in the temperature at the liquid–gas interface caused a decrease in the vapour concentration at the liquid–gas interface. Figure 4 shows a rise in the water film condensation for a higher value of Φ₀. We can justify these results by the fact that the increase in Φ₀ accelerated the cooling (Figure 3a) causing an elevation in the water vapour film condensation (Figure 4). This result is very important because the enhancement of water film condensation occurs only by the dispersion of copper nanoparticles in the liquid film and without an increase in energy consumption. Figure 5 shows the axial evolution of the relative heat fluxes for various values of Φ₀, which can be used to analyse the relative contributions of the heat exchange through sensible and latent heat. Figure 5a shows that the relative latent heat transfer was improved with the nanoparticles’ dispersion in the water, while the relative sensible heat transfer (Figure 5b) was reduced for a higher value of Φ₀. We can explain these results by the fact that that the dispersion of aluminium nanoparticles in the water improved the liquid cooling and consequently enhanced the vapour film condensation inducing an augmentation of the relative latent heat and a reduction in the relative sensible heat. Figure 6 illustrates the liquid velocity at the channel exit for various values of Φ₀. An augmentation of Φ₀ reduced the velocity of the liquid at the channel exit. These results can be explained by the fact that an augmentation of Φ₀ induced an increase in the nanofluid viscosity compared to the basic fluid (water) causing a deceleration of the liquid film flow.

4.2. Effect of Nanoparticle Types

In this work, we studied the impact of the nanoparticle type on the mass and heat transfer during liquid film condensation flowing down on a cooled vertical plate. Figure 7a shows that the gas–liquid interface temperature was less important for the water–aluminium nanofluid compared to the water–copper nanofluid and to the basic fluid. This result can be explained by the lower heat capacity and the higher thermal conductivity of the copper nanoparticles compared to the aluminium nanoparticles and to the basic fluid. Figure 8a,b present the evolution of the sensible and latent heat fluxes for different nanofluids and for pure water. The heat transfer was enhanced with the dispersion of the nanoparticles. The sensible and latent heat fluxes were much better for the water–copper nanofluid, due to the higher thermal conductivity and lower heat capacity of the copper nanoparticles compared to the aluminium nanoparticles and to pure water. Figure 9 presents the condensation rate evolution for various nanoparticle types. The liquid film condensation was higher for the copper–water nanofluid compared to the aluminium–water nanofluid and to the pure water, owing to the fact that the dispersion of the nanoparticles enhanced the latent heat flux (Figure 8a). Figure 10 displays the effect of the ambient humidity on the liquid film condensation. The liquid condensation was important for a higher value of the vapour mass fraction c₀. Figure 11 illustrates the influence of the thermal flux cooling q₁ on the film condensation. This result demonstrates that a rise in flux cooling q₁ improved the film condensation. Figure 12 shows that a rise in the mass flow rate m_0L accelerated the film concentration rate. An increase in the mass flow rate m_0L ameliorated the liquid cooling causing an enhancement in the vapour condensation. Figure 13 shows that a rise in the inlet vapour velocity u₀ accelerated the cumulated condensation, due to the fact that an increase in u₀ precluded the water vapour from being condensed on the liquid film.

5. Conclusions

The present work focused on the numerical investigation of the condensation of falling liquid film containing different types of nanoparticles with low-volume fraction. The nanofluid film fell inside a heat exchanger by mixed convection. The heat exchanger was composed of two parallel vertical plates. One of the plates was wetted and heated, while the other plate was isothermal and dry. The impact of the volume fraction and dispersion of several types of nanoparticles in the film on the heat transfer, mass transfer, and the condensation process was examined in this work.

The conclusions are as follows:

(1)

The dispersion of the nanoparticles improved the mass and heat exchange during film condensation.

(2)

The dispersion of the nanoparticles improved the film condensation.

(3)

The mass and heat exchange were improved by using the Cu–water nanofluid compared to the Al–water nanofluid.

(4)

Compared to the Al nanoparticles and pure water, the liquid film condensation was considerably improved in the case of Cu nanoparticles.

(5)

It was observed that an increase in the inlet gas mass fraction improved the water film condensation.

(6)

An increase in the thermal flux cooling improved the water film condensation.

(7)

An increase in the inlet gas velocity worsened the water film condensation.