# Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model

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## Abstract

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## 1. Introduction

## 2. Basic Thermal Conductivities Models for CNTs

## 3. Mathematical Modeling

#### 3.1. Physical Quantities of Interest

#### 3.2. Entropy Analysis and Bejan Numbers

## 4. Solution by HAM

## 5. Results and Discussion

#### 5.1. Velocity Profile

#### 5.2. Temperature Function $\theta \left(\xi \right)$

#### 5.3. Entropy Generation ($Ns$) and Bejan Number ($Be$)

#### 5.4. Tables Discussion

## 6. Conclusions

- (a)
- The velocity function ${f}^{\prime}\left(\xi \right)$ increased with the augmentation in $\phi $, positive $Q$, ${n}_{i}$, and ${n}_{e}$, while it reduced with higher values of $R$, $M$, $Kr$, and negative $Q$.
- (b)
- It is observed that the transverse velocity function $g\left(\xi \right)$ increased with greater value of $\phi ,{n}_{i},\mathrm{and}\text{}{n}_{e}$, while it showed a reducing behavior for higher values of $R$, $Kr$, and $M$.
- (c)
- The temperature function $\theta \left(\xi \right)$ was augmented with the augmentation in $\phi $, while it showed reducing behavior with the escalation in $R,Pr$.
- (d)
- For entropy profile, it was observed that entropy generation $Ns$ increased with higher value of $M,Re,\mathrm{and}\text{}Br,$ while it showed decreasing behavior with an increase in ${n}_{i}\mathrm{and}\text{}{n}_{e}$.
- (e)
- The Bejan number $Be$ showed increasing behavior with an increase in $M$, $Br$, while it showed decreasing behavior with an increase in ${n}_{i}\mathrm{and}\text{}{n}_{e}$.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Pr | Prandtl number $\left(=\upsilon /\alpha \right)$ | $\psi $ | similarity variables |

P | fluid pressure $\left(\mathrm{Pa}\right)$ | ${\tau}_{xy}$ | surface shear stress |

$Nu$ | Nusselt number | $\sigma $ | electrical conductivity |

$g$ | internal energy distribution functions | $\alpha $ | thermal diffusivity |

${B}_{0}$ | magnetic flux density $\left({\mathrm{NmA}}^{-1}\right)$ | τ | lattice relaxation time |

$Be$ | Bejan number. | $\phi $ | volume friction |

$m$ | Hall parameter | $k$ | thermal conductivity |

$M$ | magnetic parameter | $\alpha $ | thermal diffusivity $\left({\mathrm{m}}^{2}{\mathrm{s}}^{-1}\right)$ |

$J$ | current density | $\Omega $ | angular velocity $\left({\mathrm{ms}}^{-1}\right)$ |

$E$ | electric intensity | $\mu $ | dynamic viscosity $\left(\mathrm{mPa}\right)$ |

${n}_{e}$ | ion-slip parameter | $\omega $ | electron cyclotron |

a, b, c | constants | ||

$T$ | fluid temperature $K$ | Subscripts | |

${c}_{p}$ | specific heat $\left(\frac{J}{kgK}\right)$ | $nf$ | nanofluid |

$CNTs$ | carbon nanotubes | ||

${\tilde{C}}_{f}$ | skin friction coefficient | $h$ | hot |

$R$ | Reynolds number | $ave$ | average |

$Q$ | suction and injection | ||

$Kr$ | rotation parameter | ||

$h$ | distance between the plates $\left(\mathrm{m}\right)$ | ||

${Q}_{w}$ | surface heat flux | ||

${S}_{h},{S}_{R},{S}_{f}$ | Dimensional entropy generation | ||

$Ns$ | non-dimensional entropy generation | ||

$u$, $v$ $w$ | velocities components $\left({\mathrm{ms}}^{-1}\right)$ | ||

$x,y,z$ | coordinates | ||

$O$ | origin | ||

$N{u}_{x}$ | Nusselt number | ||

Greek symbols | |||

$\upsilon $ | kinematic viscosity | ||

$\rho $ | fluid density |

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**Figure 2.**Impact of $\psi $ on ${f}^{\prime}\left(\xi \right)$, when $R=0.1,Kr=0.5,{n}_{e}=0.6,Q=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 3.**Impact of $R$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,\text{}Kr=0.5,\text{}{n}_{e}=0.6,\text{}Q=0.7,\text{}{m}_{e}=0.8,\text{}M=0.9$.

**Figure 4.**Impact of $Kr$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Q=0.7,\text{}{m}_{e}=0.8,\text{}M=0.9$.

**Figure 5.**Impact of $M$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}Q=0.9$.

**Figure 6.**Impact of $\left(Q>0\right)$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 7.**Impact of $\left(Q<0\right)$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 8.**Impact of ${n}_{i}$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,M=0.8,\text{}Q=0.9$.

**Figure 9.**Impact of ${n}_{e}$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{i}=0.6,Kr=0.7,M=0.8,\text{}Q=0.9$.

**Figure 10.**Impact of $\psi $ on $g\left(\xi \right)$, when $Pr=0.4,Kr=0.2,R=0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}M=0.8$.

**Figure 11.**Impact of $R$ on $g\left(\xi \right)$, when $Pr=0.4,Kr=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,M=0.8$.

**Figure 12.**Impact of $Kr$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}M=0.8$.

**Figure 13.**Impact of $M$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}R=0.8$.

**Figure 14.**Impact of ${n}_{i}$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,M=0.6,{n}_{e}=0.7,\text{}R=0.8$.

**Figure 15.**Impact of ${n}_{e}$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,M=0.6,{n}_{i}=0.7,\text{}R=0.8$.

**Table 1.**The numerical values of skin friction ${\tilde{C}}_{f}=\frac{{\mu}_{nf}}{{\mu}_{f}}{f}^{\u2033}\left(0\right),$ when $\psi =0.01$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Q}$ | ${\mathit{n}}_{\mathit{e}}$ | ${\mathit{n}}_{\mathit{i}}$ | ${\tilde{\mathit{C}}}_{\mathit{f}}$ at $\mathit{\xi}=0$ | ${\tilde{\mathit{C}}}_{\mathit{f}}$ at $\mathit{\xi}=1$ |
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.4 | 0.5 | 0.3 | 0.2 | −0.484638 | −0.515879 |

0.3 | −0.470976 | −0.529161 | |||||

0.5 | 0.2 | −0.458474 | −0.541466 | ||||

0.4 | −0.461839 | −0.538401 | |||||

0.6 | 0.4 | −0.465886 | −0.534719 | ||||

0.7 | −0.472578 | −0.529764 | |||||

1.0 | −1.5 | −0.483196 | −0.521086 | ||||

−0.1 | −2.187070 | −1.202060 | |||||

0.1 | −1.006670 | −0.971885 | |||||

1.5 | 0.3 | −0.835078 | 0.523539 | ||||

0.4 | 0.491335 | 0.511560 | |||||

0.5 | 0.2 | 0.498996 | −0.503849 | ||||

0.6 | 0.490763 | 0.511484 | |||||

1.0 | 0.495080 | 0.507058 |

**Table 2.**The numerical values of Nusselt number $N{u}_{x}=-\left(\frac{{k}_{nf}}{{k}_{f}}\right){\phi}^{\prime}\left(0\right),$ when $\psi =0.01$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Q}$ | ${\mathit{n}}_{\mathit{e}}$ | ${\mathit{n}}_{\mathit{i}}$ | $\mathit{P}\mathit{r}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ at $\mathit{\xi}=0$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ at $\mathit{\xi}=1$ |
---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.4 | 0.5 | 0.3 | 0.2 | 7.2 | −0.001105 | 0.000884 |

0.3 | −0.001107 | 0.000886 | ||||||

0.5 | 0.2 | −0.001110 | 0.000889 | |||||

0.4 | −0.001445 | 0.001157 | ||||||

0.6 | 0.4 | −0.001779 | 0.001429 | |||||

0.7 | −0.002337 | 0.001887 | ||||||

1.0 | −1.5 | −0.002932 | 0.002347 | |||||

−0.1 | −0.000808 | 0.000995 | ||||||

0.1 | −0.002280 | 0.001443 | ||||||

1.5 | 0.3 | −0.002596 | 0.004649 | |||||

0.4 | −0.003721 | 0.004249 | ||||||

0.5 | 0.2 | −0.003404 | 0.003887 | |||||

0.6 | −0.002348 | 0.002682 | ||||||

1.0 | 7.2 | −0.001862 | 0.002126 | |||||

7.3 | −0.001862 | 0.002126 | ||||||

7.5 | −0.001862 | 0.002126 |

**Table 3.**Physical properties of carbon nanotubes (CNTs) (Xie et al. [50]).

Materials | SWCNTs | MWCNTs |
---|---|---|

Thermal Conductivity ${\mathit{k}}_{\mathit{n}\mathit{f}}$ ($W/mK$) | 3000 | 3000 |

Specific gravity (g/cm^{3}) | $0.8$ | $1.8$ |

Strength $\mathbf{\left(}GPa\mathbf{\right)}$ | 50–500 | 10–60 |

Elastic Modulus $\mathbf{\left(}TPa\mathbf{\right)}$ | 1 | 0.3–1 |

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**MDPI and ACS Style**

Feroz, N.; Shah, Z.; Islam, S.; Alzahrani, E.O.; Khan, W. Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model. *Entropy* **2019**, *21*, 52.
https://doi.org/10.3390/e21010052

**AMA Style**

Feroz N, Shah Z, Islam S, Alzahrani EO, Khan W. Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model. *Entropy*. 2019; 21(1):52.
https://doi.org/10.3390/e21010052

**Chicago/Turabian Style**

Feroz, Nosheen, Zahir Shah, Saeed Islam, Ebraheem O. Alzahrani, and Waris Khan. 2019. "Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model" *Entropy* 21, no. 1: 52.
https://doi.org/10.3390/e21010052