Mixed Convection and Entropy Generation of an Ag-Water Nanofluid in an Inclined L-Shaped Channel

Abstract

This paper investigates the mixed convection and entropy generation of an Ag-water nanofluid in an L-shaped channel fixed at an inclination angle of 30° to the horizontal axis. An isothermal heat source was positioned in the middle of the right inclined wall of the channel while the other walls were kept adiabatic. The finite volume method was used for solving the problem’s governing equations. The numerical results were obtained for a range of pertinent parameters: Reynolds number, Richardson number, aspect ratio, and the nanoparticles volume fraction. These results were Re = 50–200; Ri = 0.1, 1, 10; AR = 0.5–0.8; and φ = 0.0–0.06, respectively. The results showed that both the Reynolds and the Richardson numbers enhanced the mean Nusselt number and minimized the rate of entropy generation. It was also found that when AR. increased, the mean Nusselt number was enhanced, and the rate of entropy generation decreased. The nanoparticles volume fraction was predicted to contribute to increasing both the mean Nusselt number and the rate of entropy generation.

1. Introduction

Despite its active classification, mixed convection flow is still a useful tool of heat transfer augmentation. It provides efficient cooling (or heating) for many engineering applications, such as for building ventilation, cooling gas turbine blades, cooling electronic devices, cooling chemical reactors, emergency cooling systems of nuclear reactors, etc. Many studies have revealed that the mixed convection flow have different issues based on the geometry of the channel or the duct. However, most researchers compromise between the heat transfer enhancement and the associated pressure losses. Shah and London [1] have given an early and very good review of forced convection using regular duct geometry. In the early designs of a corrugated channel, the pressure drop reaches up to six times that of a smooth channel, while the heat transfer augmentation was aut 3.5 times [2]. Separation and reattachment of the flow in backward- and forward-facing steps were proven to give an improvement to efficient heat transfer [3]. Finned or ribbed wall surfaces are also implemented as a passive strategy to improve the mixing of flow and increase the area of heat transfer [4,5,6].

For the ventilation requirements to obtain optimal performance and space limitation, the flow in a channel can be designed into different shapes. Mezrhab et al. [7] considered the surface radiation together with natural convection in a vented T-shaped cavity. They observed that the convective heat transfer increased with cavity height. Kasaeipoor et al. [8] investigated the mixed convection of a nanofluid in a vented T-shaped cavity in the presence of a uniform magnetic field. It was assumed that the flow entered from the bottom (i.e., the vertical leg) and exited from the horizontal leg. They reported that for the dominant forced convection, the maximum heat transfer occurred when the vertical leg was very thin in comparison to the horizontal leg. However, for the dominant natural convection, the maximum heat transfer occurred when the cavity shape was mostly square. Biserni et al. [9] optimized the geometry of an H-shaped cavity intrusion into a two-dimensional conducting body with uniform heat generation using Bejan’s constructal theory [10]. They demonstrated that it was necessary to evolve a complex geometry in order to improve heat transfer performance. Marcondes and Maliska [11] addressed a numerical method for handling the inlet velocity profile for open-ended channels. Their calculations were conducted for an L-shaped channel. An L-shaped duct plays a vital role in cooling the trailing blade of a gas turbine. Shih et al. [12] discussed an L-shaped duct with the presence of pin fins in sand particles for rotating and non-rotating conditions. Pardeshi [13] investigated the L-shaped cooling passage for the trailing edge of a gas turbine blade with a combination of ribs and pin fins under rotating and non-rotating conditions. Although the L-shaped geometry has been widely investigated, the investigations are limited to closed enclosure [14,15,16].

When the heat transfer augmentation is a challenging issue, improving the properties of the working fluid can reinforce the mixed convection. Over the past two decades, nanofluids have been demonstrated as a powerful and interesting field in heat transfer enhancement. A nanofluid is the dispersal of solid nanoparticles with superior thermal conductivity into the working base fluid [17]. A nanofluid is a familiar strategy in enclosures [18], nevertheless, they are used in vented cavities [8,19,20,21,22] and channels [23,24,25]. In practice, these vented cavities and channels are parts of an enclosed loop system because the nanofluid is toxic and expensive. Therefore, it is impractical to discharge a nanofluid into the environment. The common feature among these studies (except in some situations) was that the mixed heat transfer augments with increasing volume fraction of nanoparticles. Nanofluids serve in lowering the separation and in turn minimize the secondary flow, which benefits heat transfer enhancement.

A growing interest in improving the thermodynamic performance of thermal systems has led to extensive studies of exergy utilization and entropy generation. Fluid friction and irreversible heat transfer are the main reasons causing entropy generation and hence, attenuate the available work of the system. Bejan [26,27,28] published essential studies discovering the reasons that cause the generation of entropy in applied thermal engineering using the second law of thermodynamics. Following Bejan’s conclusions from these books, several studies have been published in this topic under different engineering circumstances. An analysis of entropy generation in enclosures with and without a nanofluid can be found in several past studies [29,30,31,32,33,34,35,36,37]. These studies recorded an increase in entropy generation with an increasing ratio of inertial to viscous forces, while there was conflict between heat transfer irreversibility and the volume fraction of nanoparticles. In addition, the inclination of enclosures was found to have a notable effect on entropy generation. In some of these studies, it was found that the magnetic field could reduce entropy generation. Regarding the regular channel [38,39,40,41,42,43,44] and step channel [45,46], it was found that the Reynolds number increases the rate of the generation of entropy.

During a continuous literature survey, we did not find a study that concentrated on entropy generation in a wide variety of irregular ducts. Thus, in this paper, we present an analysis of entropy generation in a nanofluid in an L-shaped duct with a mixed convective flow. We used this geometry because it is very important and useful in cooling systems of nuclear and chemical reactors, electronic components, etc.

2. Mathematical Modelling

As shown in Figure 1, our problem was a two-dimensional L-shaped duct, where the x-axis was measured along the bottom (i.e., the horizontal leg) of the channel, while the y-axis was measured along the vertical leg of the channel. The aspect ratio of the inner side H to the outer side L is AR. The flow enters in the negative y-direction, turns in 90°, and exits from the horizontal leg. A heat source of length L₀ was fixed at the bottom of the horizontal channel at a distance X₀ from the origin. The channel was tilted with an angle of λ = 30° to the horizontal axis. Two approaches exist for modeling nanofluid heat transfer: single-phase and two-phase. In the single-phase model, a uniform volume fraction distribution is assumed for nanofluids. In other words, the viscosity and thermal conductivity of nanofluids are formulated by a volume fraction and nanoparticle size, then continuity, momentum, and energy equations are solved for nanofluids. In the two-phase model, the volume fraction distribution equation is added to other conservation equations [47,48,49,50,51,52,53]. However, most comparative studies have concluded that the two-phase models predict almost identical hydrodynamic fields to the single-phase ones, but this is not the case for thermal fields. Haghshenas Fard et al. [54] showed that the average relative error between experimental data and the numerical results based on single- and two-phase models were 16% and 8%, respectively. The discrepancy between the two models, about 8%, arose for Reynolds number ranges between 700 and 2050.

This study was carried out for a steady-state case and the flow was assumed laminar, incompressible, and Newtonian. The buoyancy force is considered based on the Boussinesq’s approximation. Nanoparticles with a volume fraction φ were uniformly dispersed within the base fluid, where they were in thermodynamic equilibrium. The dimensionless form of the mass, momentum, and energy equations are as follows:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,$$

(1)

$$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\rho }_{nf} {\nu }_f}\frac{1}{Re}\left[\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right]+\frac{{\left(\rho \beta \right)}_{nf}}{{\rho }_{nf} {\beta }_f}Ri \theta sin\lambda ,$$

(2)

$$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{{\mu }_{nf}}{{\rho }_{nf} {\nu }_f}\frac{1}{Re}\left[\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right]+\frac{{\left(\rho \beta \right)}_{nf}}{{\rho }_{nf} {\beta }_f}Ri \theta cos \lambda ,$$

(3)

$$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{\alpha }_{nf}}{{\alpha }_f}\frac{1}{Re Pr}\left[\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right],$$

(4)

with the following boundary conditions:

• At solid walls: U = V = 0;
• Inlet port: U = 0, V = 1, and $\theta =0$;
• Outlet port: V = 0, $\partial U/\partial X=0$, $\partial \theta /\partial X=0$;
• At heat source: $\theta =1$;
• At the others solid walls: $\frac{\partial \theta }{\partial n}=0$, where n is a normal vector.

$$Re=\frac{{\rho }_f{u}_c L}{{\mu }_f},Ri=\frac{gL{\beta }_f\left(T_{\mathrm{h}}-T_{\mathrm{C}}\right)}{{u_c}^2},Pr=\frac{{\mathit{\upsilon }}_{\mathrm{f}}}{{\alpha }_{\mathrm{f}}},\theta =\frac{\left(T-T_c\right)}{\left(T_h-T_c\right)},$$

where u_c is the velocity of the inlet cold fluid, which is considered as a reference velocity.

The local Nusselt number can be written as:

$$\begin{array}{c}N{u}_l=\frac{h_{nf}L}{k_f},\\ h_{nf}=\frac{q_w}{T_h-T_c},\\ k_{nf}=-\frac{q_w}{{\left(\frac{\partial T}{\partial y}\right)}_{y=0}},\\ h=-\frac{K_{nf}{(\frac{\partial T}{\partial Y})}_{Y=0}}{T_h-T_c},\\ N{u}_l=-\frac{k_{nf}}{k_f}{\left(\frac{\partial \mathsf{\theta }}{\partial \mathrm{Y}}\right)}_{\mathrm{Y}=0}.\end{array}$$

(5)

The mean Nusselt number can be expressed as:

$$N{u}_m=\frac{1}{L_0}{\int }_A^{B}N{u}_l dX.$$

(6)

The dimensional relation of entropy generation is given by:

$$S_l=\frac{k_{nf}}{T_r^2}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]+\frac{{\mu }_{nf}}{T_r}\left[2\left\{{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right\}+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right],$$

(7)

and the non-dimensional form of the above equation is:

$$\begin{array}{c}{S}_l^{*}=\frac{S_l}{S_{l,c}} S_{l,c}=\frac{K_f{\left(\mathsf{\Delta }T\right)}^2}{L^2{T_r}^2},\\ \begin{array}{l}{S}_l^{*}=\frac{k_{nf}}{k_f}\left[{\left(\frac{\partial \theta }{\partial X}\right)}^2+{\left(\frac{\partial \theta }{\partial Y}\right)}^2\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\mu }_{nf}}{{\mu }_f}\frac{Br}{\mathsf{\Omega }}\left[2\times \left\{{\left(\frac{\partial U}{\partial X}\right)}^2+{\left(\frac{\partial V}{\partial Y}\right)}^2\right\}+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right],\end{array}\end{array}$$

(8)

where

$$T_r=\frac{T_h+T_c}{2}, \mathsf{\Omega }=\frac{\mathsf{\Delta }T}{T_r}, \mathsf{\Delta }T=T_h-T_c, Br=\frac{\mu {u_c}^2}{k\mathsf{\Delta }T}.$$

(9)

The total entropy generation throughout the whole domain can be defined as:

$$S_{gen,m}={\int }_0^1{\int }_0^1{S}_l^{*}dXdY.$$

(10)

The criterion of thermal performance against the generation of entropy that was proposed in past papers [36,37] is defined here in an inverse form as:

$$\epsilon =\frac{N{u}_m}{S_{gen,m}}.$$

(11)

In this form, we get a better display to this criterion, where its maximum values correspond to the best thermal performance.

Nanofluid Relations

Table 1. Thermo-physical properties of the used nanoparticles and the base fluid.

Thermo-Physical Properties	Ag-Nanoparticle	Water (Base Fluid)
ρ (kg·m−3)	10,500	997.1
Cp (J·kg−1·K−1)	235	4179
k (W·m−1·K−1)	430	0.613
Β × 10−5 (K−1)	5.1	21
µ × 10−4 (kg·m−1·s−1)	-	8.9
α × 10−6 (m−1·s−1)	149	0.14

presents the individual thermo-physical properties of Ag-nanoparticles and water. In this study, we considered the Ag-water nanofluid with the following thermo-physical properties.

Assuming spherical Ag-nanoparticles and following Brinkman’s and Maxwell-Garnetts’ [55] formula for dynamic viscosity and thermal conductivity, respectively, the relations of the thermo-physical properties of a nanofluid can be written as [56]:

$${\mu }_{nf}={\mu }_f{\left(1-\phi \right)}^{-2.5},$$

(12)

$$k_{nf}=k_f\left[\frac{\left(k_s+2k_f\right)-2\phi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+\phi \left(k_f-k_s\right)}\right],$$

(13)

$${\left(\rho c_p\right)}_{nf}=\left(1-\phi \right){\left(\rho c_p\right)}_f+\phi {\left(\rho c_p\right)}_s,$$

(14)

$${\left(\rho \beta \right)}_{nf}=\left(1-\phi \right){\left(\rho \beta \right)}_f+\phi {\left(\rho \beta \right)}_s.$$

(15)

3. Numerical Solutions

3.1. Methodology

We used the collected grid finite volume method for the numerical solution of the governing Equations (1)–(4). The computational domain was uniformly discretized (Figure 2) using a displaced network method with the central difference scheme to discretize diffusive terms and the up-wind scheme for convective terms. The SIMPLE algorithm [57] was used for pressure–velocity linkage. The numerical efforts were achieved by an in-house-built FORTRAN computer code (FORTRAN 90). An iterative procedure was invoked with the following convergence criterion:

$$\sum _j\sum _i\sqrt{{\left[{\frac{{\delta }^{n+1}-{\delta }^n}{{\delta }^{n+1}}|}_{i,j}\right]}^2}\le {10}^{-8},$$

(16)

where $\delta$ is U, V or θ. Once the variables U, V, and θ were computed, the Nusselt number, entropy generation, and the performance criterion were calculated.

3.2. Validations

To test grid independency, we calculated the mean Nusselt number for a wide variety of grid sizes for a Richardson number Ri = 1, AR = 0.5, nanoparticles volume fraction φ = 0.04, and λ = 30° for two different Reynolds numbers (Re = 50 and 200). As presented in

Table 2. Grid independency test for a Richardson number Ri = 1, aspect ratio AR = 0.5, nanoparticles volume fraction φ = 0.04, and angle λ = 30° for two different Reynolds numbers (Re = 50 and 200).

Number of Grids	40 × 40	60 × 60	80 × 80	100 × 100	120 × 120	140 × 140	160 × 160
Re = 50	10.953	11.134	11.228	11.285	11.331	11.331	11.331
Re = 200	14.847	13.372	12.760	12.475	12.324	12.324	12.324

, the grid size of 120 × 120 was found to be the most suitable from the accuracy and time-consumption points oview, and thus we used it in this study.

This validation was further supported by making a comparison with a vented T-shaped cavity addressed by Kasaieepoor et al. [8] for a T-shaped enclosure at Re = 100 for various Ri and AR values. The results of the comparison are presented in Figure 3, which implied a very strong agreement. This improved the confidence in our built-in code.

4. Results and Discussion

The current study was achieved by varying four pertinent parameters. These were the Reynolds number Re, Richardson number Ri, aspect ratio AR, and the nanoparticles volume fraction φ. Pure water (Pr = 6.26) was considered as the base fluid and the inclination of the channel was λ = 30°, where these two values were fixed throughout the current study. The effects of the first three pertinent parameters were categorized into three subsections, while the effect of the nanoparticles volume fraction was examined inclusively within these three subsections.

4.1. Effect of the Reynolds Number

To study the effect of the Reynolds number, we fixed the Richardson number value at Ri = 1 and the aspect ratio AR at 0.5. Figure 4 presents the contours of streamlines, isotherms, and isentropic lines for different values of the Reynolds number Re = 50–200 with an interval of 50. The streamlines tended to be narrower with increasing values of Re and turned violently through the horizontal leg of the channel. The effect of the nanoparticles on the streamlines (dashed lines) became prominent at higher values of the Reynolds number, where the density of the nanofluid, which governs the inertial force, already increased with the addition of nanoparticles. Increasing the Reynolds number led to increasing the mass flow rate, hence increasing heat transfer. As such, the thermal boundary layer close to the heat source looked thinner with increasing Re and resulted in an isothermal region away from the lower inner corner. This was an indication of a larger heat transfer. Again, the isotherms of the nanofluid were noticeably distinguished with high Re values. In accordance to the isentropic lines, the source of the entropy generation was mainly due to the heat transfer irreversibility that was close to the heat source, even with higher Re values where the friction of the fluid contributed to generating entropy as shown with Re = 150 and 200.

Figure 5 presents variations of the mean Nusselt number with the nanofluid volume fraction φ for Re = 50–200. The convective heat transfer increased with the Reynolds number and this was due to increasing the mass flow rate of the fluid/nanofluid, which augmented the overall heat transfer. Increasing the value of φ enhanced the thermal conductivity and the heat capacity of the nanofluid. Thus, we observed that the mean Nusselt number Nu_m increased gradually with φ. However, the increment of Nu_m with φ at Re = 50 was faster than the other higher Re values. This refers to the relatively low effect of the inertial force against the enhancement raised by the thermal conductivity and heat capacity.

Figure 6 shows a drastic reduction of entropy generation when the Reynolds number increased from 50 to 100. By increasing Re greater than 100, the decrease of the entropy generation became slight. This reduction of entropy generation can be attributed to the decrease of the fluid friction irreversibility because of the inverse proportionality between the nanofluid friction factor and the Reynolds number. Increasing the volume fraction of nanoparticles caused an increase in entropy generation, particularly at low Re values as shown in Figure 6. We attributed this increase to the increase in the exchange of energy and to viscous forces, as a result of the irreversibility of the heat transfer and fluid friction augment.

The criterion that quantifies convective heat transfer to generated entropy (ε), which was discussed well in Ismael et al. and Chamkha et al. [36,37], is shown in Figure 7. Referring to the variations of Nu_m and S_gen,m with Re and φ, one can understand the behavior of the ε criterion which is portrayed in Figure 7, where this figure shows the improvement of the utilized thermal energy with an increasing Reynolds number where the minimum irreversibility losses could be obtained with Re = 200. The volume fraction of the nanoparticles could slightly reduce this criterion. To gain a closer view into the effect of the nanofluid, we normalized the mean Nusselt number Nu_m, the total entropy generation S_gen,m, and the ε by the their corresponding values for a pure base fluid (φ = 0). The results are gathered in Figure 8, Figure 9 and Figure 10. These figures demonstrate the reduction of the ε criterion with the nanoparticles volume fraction in low Reynolds numbers.

4.2. Effect of the Richardson Number

To investigate the effect of the Richardson number Ri, we fixed Re at 100 and AR at 0.5. Figure 11 depicts inconsiderable variations in the streamline and isotherm maps when the convection mode changed from forced (Ri = 0.1) to natural (Ri = 10), except for the isentropic lines, where a high concentration of isentropic lines was predicted close to the heat source at Ri = 10. The nanofluid (dashed) lines also demonstrated inconsiderable variations with Ri.

Figure 12 presents the variation of the mean Nusselt number for Ri = 0.1, 1, and 10 at Re = 100 and AR = 0.5. For Ri = 1, the natural and forced convections were equivalent, hence forcing the nanofluid through the horizontal leg of the channel, which can destroy the natural convective currents and hence minimize the mean Nusselt number. For the forced convection dominance condition, Ri = 0.1, the forced nanofluid would reinforce the convection and increase the mean Nusselt number. However, with the natural convection dominance condition, Ri = 10, the mean Nusselt number was at its maximum. Similarly, the total entropy generation behaved with the Richardson number as shown in Figure 13 by giving an indication of the dominance of the heat transfer irreversibility along with a high effect of the viscous force. Figure 14 demonstrates mostly constant improvement and linear drawbacks of thermal performance with increasing values of the Richardson number and the nanoparticles volume fraction, respectively. The effect of the nanoparticles volume fraction was studied in a normalized fashion as in Figure 15, Figure 16 and Figure 17. These figures show rapid heat transfer augmentation, and mostly linear entropy generation increased with φ at high Richardson numbers whereas, in general, the nanoparticles increased the thermodynamic irreversibility as shown in Figure 17.

4.3. Effect of the Aspect Ratio

The effect of the channel size was studied by varying the aspect ratio AR from 0.5 to 0.8 for Re = 100 and Ri = 1. Figure 18 shows that at AR = 0.5, when the streamlines turned 90° through the horizontal leg, they arched far away from the lower wall of the horizontal channel. When AR increased, the space available for the fluid flow decreased and thus, the streamlines became narrower and occupied most of the entire channel. Following the streamline tendency, the isotherms were intensified close to the heater element at the lower wall of the horizontal channel. The isentropic lines also concentrated violently close to the heater element.

Figure 19 shows that the mean Nusselt number was directly proportional to the aspect ratio. This was due to the augmentation of the forced convection gained by the decreasing flow area when AR increased. Hence, increasing AR reinforced the flow intensity and the transferred heat transfer; as such, the heat transfer and the fluid friction irreversibility increased accordingly. This effect is clearly visible in Figure 20, where the rate of entropy generation increased with an increase of AR. In Figure 19 and Figure 20, the nanofluid increased both the mean Nusselt number and the entropy generation at any aspect ratio. Figure 21 suggests that there was competition between the effects of AR and φ, where reducing the channel passages (i.e., increasing AR) enhanced the utility of the thermal energy with respect to the thermodynamic irreversibility—this was true up to φ ≅ 0.045. Beyond this value, the effect of increasing φ overcame the effect of AR. This can be drawn from the conversion of curves shown in Figure 21.

The normalized parameters Nu_m, S_gen,m, and ε, which are plotted in Figure 22, Figure 23 and Figure 24, respectively, show that the nanoparticles addition increased the mean Nusselt number and the rate of entropy generation and weakened the optimum utility of the mixed convection.

5. Conclusions

Mixed convection of an Ag-water nanofluid in an inclined L-shaped channel was numerically studied using the finite volume method. By keeping the inclination of the channel fixed, the Reynolds number, Richardson number, aspect ratio, and the nanoparticles volume fraction were varied, giving the following conclusions:

The Nusselt number increased with an increase of the Reynolds number Re, while entropy generation decreased drastically when Re increased from Re = 50 to 100. However, this decrease slowed down when Re increased beyond 100. When Re = 100, the convective heat transfer and the entropy generation rate were at their maxima at higher values of the Richardson number and at their minima when the value of the Richardson number equal to unity. The narrower L-shaped channel (i.e., higher aspect ratio) augmented the convective heat transfer and the rate of entropy generation. According to the ε criterion, which was defined in this study as the ratio of the mean Nusselt number to the total entropy generation rate, the best utilization of the convective heat transfer could be attained at higher values of the Reynolds and Richardson numbers. High aspect ratios also gave the best ε values, but up to a certain limit of the nanoparticles volume fraction (φ = 0.045). Beyond this limit, the lower aspect ratio gave the best ε value.

Generally, the mean Nusselt number and the rate of entropy generation increased with increasing values of the nanoparticles volume fraction while, conversely, the utility of the convective heat transfer (ε criterion) decreased with the increase of the nanoparticles volume fraction.