# Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}O

_{3}-H

_{2}O) was considered as the fluid model. Due to the sensitivity of entropy to duct diameter, mini- and microchannels with diameters of 3 mm and 0.05 mm were considered, and a laminar flow regime was assumed. The conductivity and viscosity of two different nanofluid models were examined with the help of theoretical and experimentally determined parameter values. It was shown that order of the magnitude analysis can be used for estimating entropy generation characteristics of nanofluids in mini- and microchannels. It was found that using highly viscous alumina-water nanofluid under laminar flow regime in microchannels was not desirable. Thus, there is a need for the development of low viscosity alumina-water (Al

_{2}O

_{3}-H

_{2}O) nanofluids for use in microchannels under laminar flow condition. On the other hand, Al

_{2}O

_{3}-H

_{2}O nanofluid was a superior coolant under laminar flow regime in minichannels. The presented results also indicate that flow friction and thermal irreversibility are, respectively, more significant at lower and higher tube diameters.

## 1. Introduction

_{k}> C

_{µ}, where C

_{k}is the conductivity coefficient and C

_{µ}is the viscosity coefficient [10]. Similarly, Garg et al. [11] observed that if the diameter of the pipe through which the nanofluids flow remain sun changed, the use of a nanofluidis not helpful; however, if the pipe diameter increases proportionally to the thermal conductivity of the nanofluid, replacing the base fluid with a nanofluid, with C

_{µ}< 5C

_{k}as a coolant, becomes advantageous.

## 2. Methodology

_{gen,NF}/Ṡ

_{gen,BF}˂ 1 [9].

#### 2.1. Thermal Properties of Nanofluids

_{k}is the conductivity coefficient and C

_{µ}is the viscosity coefficient. Using the data for alumina-water nanofluid from the literature [21,22,23,24,25], the variation of conductivity ratio with solid volume fraction is plotted in Figure 1. It is seen that the majority of the data were located near a straight line that corresponds to C

_{k}= 4. Therefore, the experimental value for alumina-water nanofluid thermal conductivity is given by Equation (7) with C

_{k}= 4.

_{µ}= 10 is observed.

_{k}and Cµ, where one is calculated from the Hamilton-Cross and Einstein equations for the highly dilute suspensions and the other is based on the experimental data (C

_{µ}= 10 and C

_{k}= 4). It is acceptable for the values of Cµ and C

_{k}to be between the set two limits.

#### 2.2. Microchannels

_{1}µKT∝ 10

^{9}, Z

_{2}ρ2∝ 10

^{6}, and Z

_{2}ρ

^{2}are negligible compared to Z

_{1}µKT. Therefore:

_{P}/ρ

_{BF}is always less than 1, the ratio given by Equation (12) is always more than unity. Therefore, the use of nanofluids in microchannels under laminar flow regime is not recommended.

#### 2.3. Minichannels

^{2}, K∝ 1, ṁ∝ 10

^{(−3)}, D∝ 10

^{(−3)}, ρ∝ 10

^{3}, µ∝10

^{(−3)}, q" ∝ 10

^{4}), we find Z

_{2}ρ

^{2}∝ 10

^{8}and Z

_{1}µKT ∝ 10

^{7}.It is seen that the orders of terms Z

_{1}µKT and Z

_{2}ρ

^{2}are comparable for laminar flows in a minichannel. Thus, all parameters affect the entropy generation ratio in this case, and the result depends on the operating conditions. It is then possible to assess the effectiveness of a nanofluid for different applications. In particular, the equation for heat transfer of nanofluids in a minichannel needs to be included in the analysis. Depending on the physical properties of the nanofluids and the flow conditions, the use of nanofluids in minichannels may or may not be advantageous. The parameters for calculating entropy generation are presented in Table 1.

All the input data for calculation | |
---|---|

Data | Values |

Tin | 300 K |

∆T = (Tout-Tin) | 5 K |

Length of channel | 1 m |

Base fluid (water) density | 1,000 kg/m^{3} |

Base fluid conductivity | 0.6 W/mK |

Base fluid viscosity | 0.001 N s/m^{2} |

Base fluid Cp | 4,180 kJ/kg K |

Particle (alumina) density | 3,900 kg/m^{3} |

Particle conductivity | 40 W/m K |

Particle Cp | 880 kJ/kg K |

Laminar flows | |

(Friction factor) ƒ | 64/Re |

(Heat flux) q'' | 2,500 (w/m^{2}) |

(Reynolds) Re | 1,500 |

(Nusselt) Nu | 48/11 |

^{2}); T = 300 K.

The calculation of entropy generation in microchannel | |||||
---|---|---|---|---|---|

Base Fluid | Nanofluid | ||||

Volume Fraction | 0 | 0.02 | 0.06 | 0.1 | 0.14 |

Ṡgen, thermal (kJ/kgK) | 2.083E^{-7} | 1.929E^{-7} | 1.679E^{-7} | 1.488E^{-7} | 1.3354E^{-7} |

Ṡgen, frictional (kJ/kgK) | 0.0754 | 0.116384 | 0.2241 | 0.36246 | 0.46401 |

Total (kJ/kgK) | 0.0754 | 0.116385 | 0.2241 | 0.36246 | 0.46401 |

Be (Bejan number) | 3.76E^{-7} | 2.7E^{-7} | 2.7E^{-7} | 1.92E^{-7} |

The calculations of entropy generation in minichannels | |||||
---|---|---|---|---|---|

Base Fluid | Nanofluid | ||||

Volume Fraction | 0 | 0.02 | 0.06 | 0.1 | 0.14 |

Ṡ_{gen}, thermal (kJ/kgK) | 7.4994E^{-4} | 6.944E^{-4} | 6.048E^{-4} | 5.357E^{-4} | 4.801E^{-4} |

Ṡ_{gen}, frictional (kJ/kgK) | 2.0942E^{-5} | 3.233E^{-5} | 6.224E^{-5} | 1.007E^{-4} | 1.464E^{-4} |

total (kJ/kgK) | 7.7088E^{-4} | 7.267E^{-4} | 6.67E^{-4} | 6.364E^{-4} | 6.272E^{-4} |

Be (Bejan number) | 1.256 | 1.397 | 1.594 | 1.873 |

_{genNF}/Ṡ

_{genBF}decreases with an increase in volume fraction.

- ${\text{\u1e60}}_{\mathrm{gen}\u2206T}$ = entropy generation due to thermal irreversibility and
- ${\text{\u1e60}}_{\mathrm{gen}\u2206P}$ = entropy generation due to frictional irreversibility.

## 4. Conclusions

_{2}O

_{3}-H

_{2}O) nanofluid under laminar flow regime in microchannels, it was observed that the ratio of entropy generation for the nanofluid over the base fluid is higher than unity, and the ratio increases with the increase in solid volume fraction. Therefore, the use of the alumina-water nanofluids in microchannels is not recommended. In minichannels, however, the entropy generation rate ratio is less than one and decreases with each increment in solid volume fraction. Therefore, the application of alumina-water nanofluids to a minichannel is advantageous.

## Acknowledgments

## Nomenclature

Ф | volume fraction | C_{p} | Specific heat J/KG. k |

ƒ | friction factor | K | thermal conductivity |

D | diameter of tube, m | N | shape constant |

Nu | Nusselt Number | Ρ | density |

Re | Reynolds Number | ṁ | mass flow rate |

Be | Bejan Number | µ | Viscosity |

Ṡ_{gen} | entropy generation per unit length, W/m.k | Subscript | |

$q"$ | heat flux per unit length, W/m | BF | base fluid |

C_{µ} | viscosity coefficient | NF | nanofluid |

C_{k} | thermal conductivity coefficient | P | nanoparticles |

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**MDPI and ACS Style**

Hassan, M.; Sadri, R.; Ahmadi, G.; Dahari, M.B.; Kazi, S.N.; Safaei, M.R.; Sadeghinezhad, E.
Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels. *Entropy* **2013**, *15*, 144-155.
https://doi.org/10.3390/e15010144

**AMA Style**

Hassan M, Sadri R, Ahmadi G, Dahari MB, Kazi SN, Safaei MR, Sadeghinezhad E.
Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels. *Entropy*. 2013; 15(1):144-155.
https://doi.org/10.3390/e15010144

**Chicago/Turabian Style**

Hassan, Mohammadreza, Rad Sadri, Goodarz Ahmadi, Mahidzal B. Dahari, Salim N. Kazi, Mohammad R. Safaei, and Emad Sadeghinezhad.
2013. "Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels" *Entropy* 15, no. 1: 144-155.
https://doi.org/10.3390/e15010144