## 1 Introduction

The increasing demand for energy, particularly in recent years, and the deteriorating fossil fuel resources have encouraged researchers and scientists to develop devices and equipment with improved efficiencies to reduce energy consumption [1]. These technological developments boosted power densities due to high-performance requirements. So, the conventional approaches of cooling these devices need to be modified. The modification is done in the form of size reduction to the micro- and nano-scale levels, which results in the miniaturization of these devices and improvement in the heat transfer (HT). Tuckerman and Pease [2] coined the concept of using MCHS for cooling of integrated circuits in the application of very large-scale integration (VLSI). This concept of cooling of electronics devices received much attention in the past few decades and helped in enhancing the feasibility of integrated circuits even at high power densities. The MCHS systems are cheap and cost-effective [3]. The thermal systems have different thermal boundary conditions (BCs), such as uniform wall temperature, uniform heat flux, or insulated wall BCs, depending upon the various applications [4].

A thermal system undergoing any process or processes has some irreversibility. This irreversibility results in the thermal system’s efficiency loss and can be associated with friction, mass transfer, and the thermal gradient. For convective flow, energy utilization performance is illustrated by two quantities, i. e., irreversibility and HT rate.

Newton’s law of cooling and the second law of thermodynamics are used for the prediction of the HT rate and entropy generation (EG) or exergy loss, respectively. The entropy generation rate (EGR) is the measure of the irreversibilities related to HT and viscous effects in thermal-fluid systems [5]. The thermodynamic devices and processes having thermodynamic imperfection with respect to mass transfer, HT, and other transport processes are modeled and optimized using the EG minimization method [6]. The total exergy inflow, outflow, and exergy destructed from the system are involved in the analysis of exergy [7]. The exergy analysis can be used for the systems of nano-/micro-size or nano-/micro-sections of devices [8], [9]. Furukawa and Yang [10] found that the regions where local irreversibility increases significantly can be identified by the local EG distribution and thereby it provides vital information for optimal geometry design by minimizing the EG in the system. Bejan [11] explained the

Abbassi [12] found that the MCHS shows best thermal performance corresponding to the maximum aspect ratio (AR) of the channel because thermal EGR decreases with increasing AR. Avci and Aydin [13] concluded that EG increases with increasing Brinkman number (

Guo et al. [16] observed that for the case of smaller curvature radius curved microchannels, the total

The increment in

Chauhan et al. [31] analyzed the effect of thermophysical property variations on the EG when approaching the microscale.

Chauhan et al. [33] analyzed the EG in the water flowing through the channels with the dimension approaching from the macroscale to the microscale. This optimum value was found from the plot where

The comprehensive study of the literature highlighted that most of the earlier studies are restricted to the assessment of the HT parameters and EG, whereas less attention was paid to the optimization of the number of channels and the channel diameter using numerical simulations at the microscale. The present investigation can be beneficial in applications such as cooling of electronic devices (graphics cards, integrated circuits or chips, etc.), gas turbine blades, solar collectors, etc. It may be concluded from the preceding discussion that the present study frames an innovative, interesting, and valuable study of the convective fluid flow and HT performance through the microchannels subjected to steeper temperature gradients measured in terms of EG.

Cases with parametric description.

Cases | N | Wall heat flux, ^{2}) | |

1 | 140 | 84.52 | 8452 |

2 | 160 | 79.06 | 7906 |

3 | 180 | 74.54 | 7454 |

4 | 200 | 70.72 | 7072 |

5 | 220 | 67.42 | 6742 |

6 | 224 | 66.82 | 6682 |

7 | 225 | 66.67 | 6667 |

8 | 230 | 65.94 | 6594 |

9 | 240 | 64.55 | 6455 |

10 | 260 | 62.02 | 6202 |

11 | 280 | 59.76 | 5976 |

12 | 300 | 57.74 | 5774 |

13 | 320 | 55.90 | 5590 |

14 | 400 | 50.00 | 5000 |

## 2 Optimization using entropy generation minimization

The present work pursued the following objectives: (i) optimizing the number of channels and the corresponding microchannel diameter; (ii) establishing the relationship between the microchannel diameter and *N* in m^{2}, *N* in kg/s, and *ρ*), specific heat (*k*), and dynamic viscosity (*μ*) at 50 °C for water are 988 kg/m^{3}, 4180.6 J/kg-K, 0.6435 W/m-K, and 0.0005465 Pa-s, respectively [34]. The fluid flow through the channels is single phase as the local temperature remains in the range of 273 K to 372 K. The continuum approximation is employed in this study by treating the flowing fluid as continuous in the computational domain.

The finite volume method has been used to obtain the solution of the two-dimensional governing equations for the given initial conditions and BCs for fluid flow and HT. The ANSYS WORKBENCH is used for pre-processing, while for post-processing the FLUENT solver is used. The velocity field and pressure in the radial and axial momentum equations are coupled by adopting the Semi-implicit method for pressure-linked equations (SIMPLE) algorithm. Furthermore, the standard scheme is used for the discretization of the pressure gradient. The convergence criterion for the energy equation is set as 10^{−16}.

The initial conditions and BCs for the study are as following:

### At the inlet ($z=0$ )

Fluid at the inlet is considered with inlet temperature

### At the wall ($r=R$ )

The constant wall heat flux

### At the axis ($r=0$ )

At the axis, i. e., at the center line in Figure 1, an axisymmetric BC is applied and flow parameters (

### At the outlet ($z=l$ )

The exit pressure is considered as atmospheric pressure at outlet conditions, i. e.,

Analysis based on the first law of thermodynamics is performed using the Hagen–Poiseuille equation to express the hydrodynamically and thermally fully developed velocity and temperature profiles of the forced convective laminar flow, respectively. Analysis based on the second law of thermodynamics deals with the irreversibility associated with the thermal system. The quantitative measurement of irreversibility in any process is the EG and the volumetric EGR is calculated by *ψ* (VD function) and *N* and

The minimum value of

The model precision is ensured by performing a grid independence test for five different grid systems. The test is performed for the water (coolant) flow through the microchannel with diameter 66.67 μm. The considered grid systems are 50×125, 100×250, 160×400, 200×500, and 240×600.

## 3 Results and discussion

The present study aims to numerically analyze the second law for the laminar forced convective flow of water through the microchannels. With an increasing number of channels (corresponding to a reduction in diameter), *N*. However, for the selected microchannel geometry range, a decrease in *N* increases. With augmentation in *N*, the respective increase and decrease in

The *N*, i. e.,

The plot between *N* is shown in Figure 3, which illustrates that with an increasing number of channels,

Corresponding to the optimum value of the Bejan number, the optimum value of the diameter (

*N* is shown in Figure 5. The

The exergy loss (*N*, and the lowest exergy loss is 31.48 mW, for

A mathematical model is used to validate the results obtained from the numerical simulation. An explicit mathematical model is implemented to calculate

*h*is calculated using its correlation with the Nusselt number (

Figure 7 shows the comparison between the results obtained from the numerical simulation and from the explicit method calculation in the form of

## 4 Conclusions

The second-law analysis is performed numerically on the water flowing through the modeled two-dimensional circular MCHS subjected to a constant wall heat flux BC. The study concludes that:

- 1.With increasing or decreasing number of channels corresponding to the diameter, change occur in the values of the
and${S}_{gen,Fr}$ , while${S}_{gen,HT\text{-}rad}$ remains almost constant for the cases under consideration.${S}_{gen,HT\text{-}ax}$ - 2.The velocity gradient in the radial direction increases as the number of channels is augmented. This results in the supremacy of
over${S}_{gen,Fr}$ at higher numbers of channels corresponding to the diameter.${S}_{gen,HT}$ - 3.The
decreases with increasing number of channels as a result of a decrement in the temperature gradient in the radial direction.${S}_{gen,HT}$ - 4.The optimum value of the number of channels corresponding to the optimum microchannel diameter exists where the sum of
and${S}_{gen,Fr}$ is minimum (i. e.,${S}_{gen,HT}$ is minimum). The value of${S}_{gen,total}$ is 224.${N}^{\ast}$ - 5.The Bejan number (
) corresponding to$B{e}^{\ast}$ is 0.50, which indicates that the contribution of${N}^{\ast}$ to${S}_{gen,HT}$ is 50 % at${S}_{gen,total}$ .${N}^{\ast}$ - 6.The optimum diameter of the microchannel is 66.82 μm, corresponding to the optimum value of the Bejan number (i. e.,
).$B{e}^{\ast}=0.50$ - 7.The EG number corresponding to
is 0.1532.${N}^{\ast}$ - 8.The exergy loss is lowest for the 224 number of channels, and its value is 31.48 mW.
- 9.The minimum EG number for the explicit mathematical model is obtained for the same number of channels as in the case of the numerical method, and the error between the values of
and${N}_{S\text{-}\mathrm{Explicit}}$ is 0.648 %.${N}_{S\text{-}\mathrm{Numerical}}$

The authors are very thankful to Mr. Prathvi Raj Chauhan, research scholar at the Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India, for his valuable discussions and suggestions.

^{}

**Conflict of interest:** No conflict of interest.

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