# Natural Convection and Entropy Generation in a Square Cavity with Variable Temperature Side Walls Filled with a Nanofluid: Buongiorno’s Mathematical Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{6}. It is also important to point out the very recently published paper by Kefayati et al. [34] on simulation of natural convection and entropy generation of non-Newtonian nanofluid in an inclined cavity using Buongiorno’s mathematical model (Part II, entropy generation). The review paper on entropy generation in nanofluids flow by Mahian et al. [35] should also be mentioned in this context.

## 2. Basic Equations

_{c}. The left vertical wall have variable temperature distribution according to the vertical coordinate as follows:

_{h}> T

_{c}, ε is amplitude of the variable hot-wall temperature and κ is wave number of the hot-wall temperature.

**V**is the velocity vector, T is the nanofluid temperature, C is the nanoparticle volume fraction, t is the time, p is the fluid pressure,

**g**is the gravity vector, D

_{B}is the Brownian diffusion coefficient, D

_{T}is the thermophoretic diffusion coefficient, ${j}_{p}=-{\rho}_{p}\left[{D}_{B}\text{\hspace{0.17em}}\nabla C+\left({D}_{T}/{T}_{c}\right)\text{\hspace{0.17em}}\nabla T\right]$ is the nanoparticles mass flux, ${\rho}_{f0}$ is the reference density of the fluid, $\alpha ,\mu ,{\rho}_{p}$ denote the thermal diffusivity of the nanofluid, the dynamic viscosity, nanoparticle mass density, respectively, δ is a quantity defined by $\delta ={\left(\rho \text{\hspace{0.17em}}{C}_{p}\right)}_{p}/{\left(\rho \text{\hspace{0.17em}}{C}_{p}\right)}_{f}$, ${C}_{p}$ is the heat capacity at constant pressure, ${\left(\rho {C}_{p}\right)}_{f}$ is heat capacity of the base fluid, ${\left(\rho {C}_{p}\right)}_{p}$ is effective heat capacity of the nanoparticle material and β is the coefficient of thermal expansion.

## 3. Numerical Method and Validation

^{−7}.

_{1}= 0.0001, χ

_{2}= 0.5, χ

_{3}= 0.01. Values of the irreversibility factors are the same with those used in the paper by Magherbi et al. [48]. Six cases of the uniform grid are tested: a grid of 100 × 100 points, a grid of 150 × 150 points, a grid of 200 × 200, a grid of 250 × 250 points, a grid of 300 × 300 points, and a grid of 400 × 400 points. Table 1 shows the effect of the mesh parameters on the average Nusselt number on the hot wall. On the basis of the conducted verifications the uniform grid of 200 × 200 points has been selected for the following analysis.

## 4. Results and Discussion

^{3}–10

^{5}), Prandtl number (Pr = 7.0), Lewis number (Le = 1000), buoyancy-ratio parameter (Nr = 0.1), Brownian motion parameter (Nb = 0.1), thermophoresis parameter (Nt = 0.1), amplitude of the variable hot-wall temperature (ε = 0.0–1.0), wave number of the hot-wall temperature (κ = 0.0–5.0). Particular efforts have been focused on the effects of these parameters on the fluid flow, heat transfer and entropy generation inside the cavity. Streamlines, isotherms, nanoparticles volume fraction, entropy generation parameters as well as the average Nusselt number, fluid flow rate, average Bejan number and average entropy generation for different values of the key parameters mentioned above are illustrated in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

^{3}(Figure 5a) one convective cell is formed inside the cavity illustrating an ascending flow near the left hot wall and descending flows close to the right cold wall. Distribution of temperature along the left wall has two local maximum points due to the value of the wave number, κ = 2.0. At the same time, the low value of Rayleigh number shows that heat conduction dominates near the left wall, the isotherms being uniformly distributed in these zones. Domination of heat conduction reflects a significant effect of thermophoresis inside the cavity. Therefore, one can find non-uniform distribution of nanoparticles inside the cavity, where significant reduction of φ occurs along the left hot wall and a growth of φ occurs along the right cold wall. The distribution of local entropy generation illustrates an intensification of S

_{gen}along the left wall due to significant heat transfer part (high values of temperature gradient along this wall).

^{3}. At the same time, S

_{gen}increases significantly along the vertical walls due to intensification of the convective flow and as a result a growth of temperature gradient appears.

^{3}, κ = 2.0 and different values of left wall temperature amplitude are presented in Figure 7. For the considered range of ε one can find non-significant changes in streamlines, while other parameters change significantly. High values of amplitude characterize more significant heating of the cavity due to an increase in temperature along the left vertical wall. Such heating leads to non-uniform distributions of nanoparticles inside the cavity and local entropy generation increases within the enclosure, mainly, close to the vertical walls.

_{avg}with ε is more significant for high values of Rayleigh number. One can find also that fluid flow rate is an increasing function of wall temperature amplitude and Rayleigh number. Intensity of convective flow raises more significant for high values of Rayleigh number (see Figure 9b).

_{avg}decreases with ε for low value of Rayleigh number (Ra = 10

^{3}) and increases with ε for high value of Rayleigh number (Ra = 10

^{5}). Such behavior has been mentioned above, where high values of Ra reflect more intensive convective flow and heat transfer with high values of temperature gradient. The behavior of average entropy generation is similar to average Nusselt number in the case of more significant growth for high Ra.

^{3}, ε = 1.0. It should be noted that a growth of κ illustrates an increase in heated zones along the left vertical wall that is presented in distributions of temperature. An increase in wave number does not lead to significant modifications of streamlines, while temperature, nanoparticles volume fraction and entropy generation fields change significantly. A growth of κ leads to more uniform heating of the left vertical wall and one can find more significant heating near this wall. Nanoparticles isoconcentrations become more uniform for high values of wave number. Local entropy generation increases also with κ and significant growth occurs close the left vertical wall.

^{3}) illustrates a decrease in average Bejan number with wave number due to a reduction of the heat transfer irreversibility for low intesive flow and heat transfer in comaprison with fluid friction irreversibility and mass transfer irreversibility. At the same time, the average entropy generation significantly increases with transition between constant temperature (κ = 0) and variable temperature distribution (κ = 1) for Ra = 10

^{5}due to intesification of convective flow. Average Bejan number for high Rayleigh number characterizes a weak changes owing to significant role of heat transfer irreversibility.

^{3}and Ra = 10

^{5}in the following form:

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**Streamlines ψ, isotherms θ, nanoparticles volume fraction φ and total entropy generation S

_{gen}for ε = 1.0, κ = 2.0: (

**a**) Ra = 10

^{3}, (

**b**) Ra = 10

^{5}.

**Figure 6.**Profiles of local Nusselt number along left vertical wall for ε = 1.0, κ = 2.0 and different Rayleigh numbers.

**Figure 7.**Streamlines ψ, isotherms θ, nanoparticles volume fraction φ and total entropy generation S

_{gen}for Ra = 10

^{3}, κ = 2.0: (

**a**) ε = 0.2, (

**b**) ε = 0.6, (

**c**) ε = 0.8.

**Figure 8.**Profiles of local Nusselt number along left vertical wall for Ra = 10

^{3}, κ = 2.0 and different values of amplitude.

**Figure 9.**Variations of average Nusselt number (

**a**) and fluid flow rate (

**b**) for κ = 2.0 and different values of amplitude ε and Rayleigh number.

**Figure 10.**Variations of average Bejan number (

**a**) and average entropy generation (

**b**) for κ = 2.0 and different values of amplitude ε and Rayleigh number.

**Figure 11.**Streamlines ψ, isotherms θ, nanoparticles volume fraction φ and total entropy generation S

_{gen}for Ra = 10

^{3}, ε = 1.0: (

**a**) κ = 0.0, (

**b**) κ = 1.0, (

**c**) κ = 3.0, (

**d**) κ = 5.0.

**Figure 12.**Profiles of local Nusselt number along left vertical wall for Ra = 10

^{3}, ε = 1.0 and different values of wave number.

**Figure 13.**Variations of average Nusselt number (

**a**) and fluid flow rate (

**b**) for ε = 1.0 and different values of wave number and Rayleigh number.

**Figure 14.**Variations of average Bejan number (

**a**) and average entropy generation (

**b**) for ε = 1.0 and different values of wavy number and Rayleigh number.

Uniform Grids | $\mathit{N}{\mathit{u}}_{\mathit{a}\mathit{v}\mathit{g}}$ | $\mathbf{\Delta}=\frac{\left|\mathit{N}{\mathit{u}}_{\mathit{i}\times \mathit{j}}^{\mathit{a}\mathit{v}\mathit{g}}-\mathit{N}{\mathit{u}}_{200\times 200}^{\mathit{a}\mathit{v}\mathit{g}}\right|}{\mathit{N}{\mathit{u}}_{\mathit{i}\times \mathit{j}}^{\mathit{a}\mathit{v}\mathit{g}}}\times 100\mathbf{\%}$ |
---|---|---|

100 × 100 | 4.60418 | 0.35 |

150 × 150 | 4.61435 | 0.13 |

200 × 200 | 4.62035 | – |

250 × 250 | 4.62548 | 0.11 |

300 × 300 | 4.62991 | 0.21 |

400 × 400 | 4.63827 | 0.39 |

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**MDPI and ACS Style**

Sheremet, M.A.; Grosan, T.; Pop, I.
Natural Convection and Entropy Generation in a Square Cavity with Variable Temperature Side Walls Filled with a Nanofluid: Buongiorno’s Mathematical Model. *Entropy* **2017**, *19*, 337.
https://doi.org/10.3390/e19070337

**AMA Style**

Sheremet MA, Grosan T, Pop I.
Natural Convection and Entropy Generation in a Square Cavity with Variable Temperature Side Walls Filled with a Nanofluid: Buongiorno’s Mathematical Model. *Entropy*. 2017; 19(7):337.
https://doi.org/10.3390/e19070337

**Chicago/Turabian Style**

Sheremet, Mikhail A., Teodor Grosan, and Ioan Pop.
2017. "Natural Convection and Entropy Generation in a Square Cavity with Variable Temperature Side Walls Filled with a Nanofluid: Buongiorno’s Mathematical Model" *Entropy* 19, no. 7: 337.
https://doi.org/10.3390/e19070337