Circular Microchannel Heat Sink Optimization Using Entropy Generation Minimization Method

Krishan Kumar 1 , Rajan Kumarhttp://orcid.org/https://orcid.org/0000-0002-9929-5915 1 ,  and Rabinder Singh Bharj 1
  • 1 Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India
Krishan Kumar
  • Department of Mechanical Engineering, 29755Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, Punjab, 144011, India
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Rajan KumarORCID iD: https://orcid.org/0000-0002-9929-5915 and Rabinder Singh Bharj
  • Department of Mechanical Engineering, 29755Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, Punjab, 144011, India
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

The performance of the microchannel heat sink (MCHS) in electronic applications needs to be optimized corresponding to the number of channels (N). In this study optimization of the number of channels corresponding to the diameter of the microchannel (DN) using an entropy generation minimization approach is achieved for the MCHS used in electronic applications. The numerical study is performed for constant total heat flow rate (q˙tot) and total mass flow rate (m˙tot). The results indicate that the dominance of frictional entropy generation (Sgen,Fr) increases with the reduction in diameter. However, the entropy generation due to heat transfer (Sgen,HT) decreases with the reduction in diameter. Therefore, the optimum diameter (D) is calculated corresponding to the minimum total entropy generation (Sgen,total) for the optimum number of channels (N). Furthermore, the entropy generation number (NS) and Bejan number (Be) are also calculated.

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 Sgen,total in a control volume as the sum of EG due to HT and fluid friction, i. e., Sgen,total=Sgen,HT+Sgen,Fr.

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 (Br), while it decreases with increasing Knudsen number (Kn). Hung [14] developed a mathematical model to study the effect of viscous dissipation (VD) on the EG and found that when the VD effect was incorporated in the study, the temperature distribution was found as a strong function of Br, due to which EGs also relate to Br. The total EGR in the microchannel decreases because of the decrease in the fluid friction and increase in the HT under slip flow and temperature jump conditions [15].

Guo et al. [16] observed that for the case of smaller curvature radius curved microchannels, the total NS extremum occurs earlier. The temperature distribution was affected significantly with the inclusion of the VD effect, which thereby affects the EG associated with the HT, whereas no significant influence was analyzed in the case of fluid friction irreversibility [17]. Ibáñez et al. [18] derived the optimum value of slip length which maximizes the HT. The minimum EG gives the optimum value of both the wall to fluid thermal conductivity ratio and the slip flow and it decreases with the wall heat flux [19]. Prabhu and Mahulikar [20] found that thermal conductivity variations in the axial direction cause axial conduction and significantly increase the temperature, whereas variations in the radial direction cause flattening of the temperature profile. Ebrahimi et al. [21] analyzed increment in Sgen,Fr for the case of nanofluid flow as a result of higher fluid viscosity at higher Reynold number (Re) due to lower bulk temperature of the coolant. The increase or decrease in the irreversibility is dependent on the value of Kn under uniform wall heat flux BC corresponding to an increase or decrease in the channel AR [22]. Cruz-Duarte et al. [23] found the use of nanofluid in the laminar flow environment helpful in minimizing the entropy production of the system under the condition when the volume flow rate was considered as a design variable.

The increment in Re of nanofluid flow in the MCHS resulted in the increase in the frictional EGR [24], [25]. Rastogi and Mahulikar [26] performed optimization of a circular microtube configuration using EG minimization and the optimum diameter is obtained corresponding to the minimum total EGR change, i. e., where the optimum channel diameter decreases with decreasing channel length. Rastogi and Mahulikar [27] concluded from the study that towards the microscale, the change in energy degraded into heat due to fluid friction and fluid conduction HT increases and decreases, respectively. Another study by Rastogi and Mahulikar [28] concluded that the optimum dimension of the channel occurs corresponding to the minimum Sgen,total. The increase in the volume fraction of the nanoparticle results in Sgen,Fr increment [25], [29]. Hosseini and Sheikholeslami [30] concluded that increment in the AR from 1 to 7.5 for the microchannel resulted in an upward trend of the dimensionless total NS.

Chauhan et al. [31] analyzed the effect of thermophysical property variations on the EG when approaching the microscale. Sgen,total was found to be lower in the case of constant property solution than in the cases with property variation at the microscale. Khlifi et al. [32] analytically found that for low heat flux (because frictional irreversibility dominates) it is more convenient to use water than nanofluid, whereas at high heat flux nanofluids were more efficient than water (because of HT irreversibility dominance).

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 Sgen,total is minimum. The minimum Sgen,total occurred between the two studied diameters 100 μm and 50 μm and from the plot slope was zero in between diameters 50 μm and 70 μm.

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.

Figure 1
Figure 1

Schematic diagram of the microchannel.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

Table 1

Cases with parametric description.

CasesNDN (μm)Wall heat flux, qw,N (W/m2)
114084.528452
216079.067906
318074.547454
420070.727072
522067.426742
622466.826682
722566.676667
823065.946594
924064.556455
1026062.026202
1128059.765976
1230057.745774
1332055.905590
1440050.005000

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 Sgen,total in terms of the EG number; and (iii) comparing the results obtained using numerically and explicitly. In this study, 14 circular cross-sectional microchannels of different diameter DN or radius RN are subjected to their corresponding BCs. The two-dimensional computational domain in Figure 1 is simplified considering axis symmetry conditions. Thermally as well as hydrodynamically fully developed forced convection of Hagen–Poiseuille fluid (water) flow through the fixed-length (l=100000 μm) microchannel is considered for all the cases mentioned in Table 1. The total mass flow rate (m˙tot=N.m˙N=ρ·Acs,N·um·N=0.000180101 kg/s) and total heat flow (q˙tot=31.415926 W) are fixed for all cases. Here Acs,N is the cross-sectional area of the microchannel corresponding to the N in m2, m˙N is the mass flow rate corresponding to the N in kg/s, and um is the mean velocity in m/s. The thermo-physical properties density (ρ), specific heat (cp), thermal conductivity (k), and dynamic viscosity (μ) at 50 °C for water are 988 kg/m3, 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 T0,in=273 K and without the entrance effect associated with velocity and temperature profiles. These profiles are under no-slip and no-temperature jump conditions at the wall. The radial velocity is taken as zero (v=0 m/s).

At the wall (r=R)

The constant wall heat flux [qw,N=q˙totN·(π·DN·l)] BC is adopted corresponding to the case taken into consideration. No-slip (uw=0) and no normal flow (vw=0) conditions are considered.

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 (p,T,u) differentiability at the axis is given as p/r=T/r=u/r=0.

At the outlet (z=l)

The exit pressure is considered as atmospheric pressure at outlet conditions, i. e., Pexit=Patm=1.01325 bar. The Neumann BCs on the transport variables (u,v,T) give their normal gradients as u/z=v/z=T/z=0.

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 S˙gen=S˙gen,FR+S˙gen,HT=S˙gen=μTψ+kT2(T)2. The local Sgen,total can be obtained by discretizing the ψ (VD function) and T terms in the volumetric EG using a first-order forward difference approach and then multiplying it with N and VN (volume of the channel corresponding to the N=Acs,N×l):

Sgen,total=μTi,jui,j+1ui,jri,j+1ri,j2+kTi,j2Ti+1,jTi,jzi+1,jzi,j2+kTi,j2Ti+1,jTi,jri+1,jri,j2·N·VN.

The minimum value of Sgen,total obtained by the criterion [(Sgen,total)/N]=0 gives the optimum number of the channels (N) corresponding to the optimum dimension of microchannels (D). For the optimum design of any thermo-fluidic system, it is important to have a clear understanding of the Sgen,HT to Sgen,total ratio. The maximum exergetic efficiency can be attained corresponding to this optimum design. The dimensionless number used to express this optimum design is the Bejan number (Be) and is given as the ratio of Sgen,HT to Sgen,total [35].

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), Sgen,Fr increases while Sgen,HT decreases. The increase in Sgen,Fr is because of an increasing velocity gradient and (ui,j+1ui,jri,j+1ri,j) and decreasing Ti,j with increasing N. However, for the selected microchannel geometry range, a decrease in Sgen,HT at the microscale is attributed to the decreasing Sgen,HT-rad and almost constant Sgen,HT-ax. The decrease in Sgen,HT-rad at the microscale is because of the dominance of decrease in the temperature gradient (Ti+1,jTi,jri+1,jri,j) over the decrease in Ti,j as N increases. With augmentation in N, the respective increase and decrease in Sgen,Fr and Sgen,HT result in the first decrease in Sgen,total up to a particular diameter and thereafter increase it.

Figure 2
Figure 2

The variation of Sgen,Fr, Sgen,HT, and Sgen,total with N corresponding to DN.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

The N corresponding to the D can be obtained from the maxima/minima condition [(Sgen,total)/N]=0, which exists corresponding to the minimum Sgen,total. The Sgen,Fr is equal to Sgen,HT at the point where the curves intersect each other and the number of channels corresponding to the intersection point is denoted by Nint. As Figure 2 shows, the optimum number of channels and the intersection point of the Sgen,Fr and Sgen,HT curves (Nint) lie at the same value for N, i. e., N=Nint=224.

The plot between Be and N is shown in Figure 3, which illustrates that with an increasing number of channels, Be decreases as a result of the decrease in Sgen,HT. The optimum value of the Bejan number (Be) corresponding to N is 0.50.

Figure 3
Figure 3

The variation of Be with N corresponding to DN.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

Figure 4
Figure 4

The variation of DN with Be.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

Corresponding to the optimum value of the Bejan number, the optimum value of the diameter (D) is obtained as 66.82 μm (as in Figure 4). The correlation between Be and DN is obtained as non-linear with a coefficient of determination of 0.999:

D/L=0.0003845+0.00040Be+0.00033Be2,
NS=0.49751003×(DN/l)+729955×(DN/l)2.

Sgen,total can be calculated by its correlation with DN given in eq. (3) for the specified conditions as well as the geometry of the microchannel. The coefficient of determination is 0.98. The NS is also calculated by using the formulation NS=Sgen,total/(m˙tot.cp) [36] and the minimum value (NS,minimum=0.1532) is obtained corresponding to the optimum number of channels. The plot between NS and N is shown in Figure 5. The NS reduces as the number of channels increases from 140 onward and attains a minimum value corresponding to N=224 and thereafter it increases with increasing number of channels. This is because of the decrease and increase in Sgen,total as the number of channels increases.

Figure 5
Figure 5

The variation of the entropy generation number (NS) with N corresponding to DN.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

The exergy loss (Xloss) in the flow is the result of the irreversibility due to HT and fluid flow and is the degree of the thermodynamic efficiency of the process. Lower Xloss corresponds to higher thermodynamic efficiency [37]. The evaluation of Xloss is done using the formula Xloss=T0,in×Sgen,total. Figure 6 shows the plot for Xloss and N, and the lowest exergy loss is 31.48 mW, for N=224.

Figure 6
Figure 6

The variation of exergy loss (Xloss) with N corresponding to DN.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

A mathematical model is used to validate the results obtained from the numerical simulation. An explicit mathematical model is implemented to calculate Sgen,total as follows [38], [39]:

Sgen,total=m˙tot·cpln1+BT0,in·l1+qw,Nh·T0,in1+qw,Nh·T0,in+BT0,in·l+f8·ρ·um3qw,N·ln1+BT0,in·l·103,
where B=4·qw,Nρ·um·cpDN, the friction factor for laminar flow f=64Re=64·μρ·um·Dh, and h is calculated using its correlation with the Nusselt number (Nu) for laminar fully developed flow of water (as cold fluid) through microscale tubes with constant wall heat flux BC as follows [40]:
Nu=h.Dhk=4.364+0.01·Gz1.71+0.01·Gz1.3·1,
where the Graetz number (Gz)=Re·Pr·DNl and the Prandtl number (Pr)=μ·cpk.

Figure 7 shows the comparison between the results obtained from the numerical simulation and from the explicit method calculation in the form of NS, and the deviation among the two is very small. The maximum deviation from the explicit method is 1.75 %, for the highest number of channels.

Figure 7
Figure 7

The variation of the entropy generation number (NS), with N corresponding to DN for the numerical and explicit methods.

Citation: Journal of Non-Equilibrium Thermodynamics 45, 4; 10.1515/jnet-2019-0086

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. 1.With increasing or decreasing number of channels corresponding to the diameter, change occur in the values of the Sgen,Fr and Sgen,HT-rad, while Sgen,HT-ax remains almost constant for the cases under consideration.
  2. 2.The velocity gradient in the radial direction increases as the number of channels is augmented. This results in the supremacy of Sgen,Fr over Sgen,HT at higher numbers of channels corresponding to the diameter.
  3. 3.The Sgen,HT decreases with increasing number of channels as a result of a decrement in the temperature gradient in the radial direction.
  4. 4.The optimum value of the number of channels corresponding to the optimum microchannel diameter exists where the sum of Sgen,Fr and Sgen,HT is minimum (i. e., Sgen,total is minimum). The value of N is 224.
  5. 5.The Bejan number (Be) corresponding to N is 0.50, which indicates that the contribution of Sgen,HT to Sgen,total is 50 % at N.
  6. 6.The optimum diameter of the microchannel is 66.82 μm, corresponding to the optimum value of the Bejan number (i. e., Be=0.50).
  7. 7.The EG number corresponding to N is 0.1532.
  8. 8.The exergy loss is lowest for the 224 number of channels, and its value is 31.48 mW.
  9. 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 NS-Explicit and NS-Numerical is 0.648 %.

Acknowledgment

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.

References

  • [1]

    Z. Li, A. Shahsavar, K. Niazi, A. A. A. A. Al-Rashed and P. Talebizadehsardari, The Effects of Vertical and Horizontal Sources on Heat Transfer and Entropy Generation in an Inclined Triangular Enclosure Filled with Non-Newtonian Fluid and Subjected to Magnetic Field, Powder Technol. 364 (2020), 924–942.

    • Crossref
    • Export Citation
  • [2]

    D. B. Tuckerman and R. F. W. Pease, High-performance heat sinking for VLSI, IEEE Electron Device Lett. 2 (1981), no. 5, 126–129.

    • Crossref
    • Export Citation
  • [3]

    Z. X. Li, U. Khaled, A. A. Al-Rashed, M. Goodarzi, M. M. Sarafraz and R. Meer, Heat Transfer Evaluation of a Micro Heat Exchanger Cooling with Spherical Carbon-Acetone Nanofluid, Int. J. Heat Mass Transf. 149 (2020), 119124.

  • [4]

    R. Sarma, A. J. Nath, T. Konwar, P. K. Mondal and S. Wongwises, Thermo-Hydrodynamics of a Viscoelastic Fluid Under Asymmetrical Heating, Int. J. Heat Mass Transf. 125 (2018), 515–524.

    • Crossref
    • Export Citation
  • [5]

    M. Saffaripour and R. Culham, Measurement of Entropy Generation in Microscale Thermal-Fluid Systems, J. Heat Transf. 132 (2010), no. 12, 121401.

  • [6]

    L. Chen, C. Wu and F. Sun, Finite Time Thermodynamic Optimization or Entropy Generation Minimization of Energy Systems, J. Non-Equilib. Thermodyn. 24 (1999), no. 4, 327–359.

  • [7]

    Z. X. Li, A. Shahsavar, A. A. Al-Rashed, R. Kalbasi, M. Afrand and P. Talebizadehsardari, Multi-Objective Energy and Exergy Optimization of Different Configurations of Hybrid Earth-Air Heat Exchanger and Building Integrated Photovoltaic/Thermal System, Energy Convers. Manag. 195 (2019), 1098–1110.

    • Crossref
    • Export Citation
  • [8]

    M. A. Rosen, Size Considerations in Applications of Exergy Analysis, Int. J. Exergy 5 (2008), no. 3, 249–274.

    • Crossref
    • Export Citation
  • [9]

    K. Kumar, R. Kumar and R. S. Bharj, Entropy Generation Analysis due to Heat Transfer and Nanofluid Flow Through Microchannels: A Review, Int. J. Exergy 31 (2020), no. 1, 49–86.

    • Crossref
    • Export Citation
  • [10]

    T. Furukawa and W. J. Yang, Thermal Optimization of Channel Flows with Discrete Heating Sections, J. Non-Equilib. Thermodyn. 28 (2003), no. 4, 299–310.

  • [11]

    A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, 1995.

  • [12]

    H. Abbassi, Entropy Generation Analysis in a Uniformly Heated Microchannel Heat Sink, Energy 32 (2007), no. 10, 1932–1947.

    • Crossref
    • Export Citation
  • [13]

    M. Avci and O. Aydin, Second-Law Analysis of Heat and Fluid Flow in Microscale Geometries, Int. J. Exergy 4 (2007), no. 3, 286.

    • Crossref
    • Export Citation
  • [14]

    Y. M. Hung, A Comparative Study of Viscous Dissipation Effect on Entropy Generation in Single-Phase Liquid Flow in Microchannels, Int. J. Therm. Sci. 48 (2009), no. 5, 1026–1035.

    • Crossref
    • Export Citation
  • [15]

    W. A. Khan, J. R. Culham and M. M. Yovanovich, Optimization of Microchannel Heat Sinks Using Entropy Generation Minimization Method, IEEE Trans. Compon. Packag. Technol. 32 (2009), 243–251.

    • Crossref
    • Export Citation
  • [16]

    J. Guo, M. Xu, J. Cai and X. Huai, Viscous Dissipation Effect on Entropy Generation in Curved Square Microchannels, Energy 36 (2011), no. 8, 5416–5423.

    • Crossref
    • Export Citation
  • [17]

    W. H. Mah, Y. M. Hung and N. Guo, Entropy Generation of Viscous Dissipative Nanofluid Flow in Microchannels, Int. J. Heat Mass Transf. 55 (2012), no. 15-16, 4169–4182.

    • Crossref
    • Export Citation
  • [18]

    G. Ibáñez, A. López, J. Pantoja, J. Moreira and J. A. Reyes, Optimum Slip Flow Based on the Minimization of Entropy Generation in Parallel Plate Microchannels, Energy 50 (2013), 143–149.

    • Crossref
    • Export Citation
  • [19]

    G. Ibáñez, A. López, J. Pantoja and J. Moreira, Combined Effects of Uniform Heat Flux Boundary Conditions and Hydrodynamic Slip on Entropy Generation in a Microchannel, Int. J. Heat Mass Transf. 73 (2014), 201–206.

    • Crossref
    • Export Citation
  • [20]

    S. V. Prabhu and S. P. Mahulikar, Effects of Density and Thermal Conductivity Variations on Entropy Generation in Gas Micro-Flows, Int. J. Heat Mass Transf. 79 (2014), 472–485.

    • Crossref
    • Export Citation
  • [21]

    A. Ebrahimi, F. Rikhtegar, A. Sabaghan and E. Roohi, Heat Transfer and Entropy Generation in a Microchannel with Longitudinal Vortex Generators Using Nanofluids, Energy 101 (2016), 190–201.

    • Crossref
    • Export Citation
  • [22]

    L. Kuddusi, Entropy Generation in Rectangular Microchannels, Int. J. Exergy 19 (2016), no. 1, 110.

    • Crossref
    • Export Citation
  • [23]

    J. M. Cruz-Duarte, A. Garcia-Perez, I. M. Amaya-Contreras, C. R. Correa-Cely, R. J. Romero-Troncoso and J. G. Avina-Cervantes, Design of Microelectronic Cooling Systems Using a Thermodynamic Optimization Strategy Based on Cuckoo Search, IEEE Trans. Compon. Packag. Manuf. Technol. 7 (2017), no. 11, 1804–1812.

    • Crossref
    • Export Citation
  • [24]

    E. Manay, E. F. Akyürek and B. Sahin, Entropy Generation of Nanofluid Flow in a Microchannel Heat Sink, Results Phys. 9 (2018), 615–624.

    • Crossref
    • Export Citation
  • [25]

    A. A. A. A. Al-Rashed, A. Shahsavar, O. Rasooli, M. A. Moghimi, A. Karimipour and M. D. Tran, Numerical Assessment into the Hydrothermal and Entropy Generation Characteristics of Biological Water-Silver Nano-Fluid in a Wavy Walled Microchannel Heat Sink, Int. Commun. Heat Mass Transf. 104 (2019), 118–126.

    • Crossref
    • Export Citation
  • [26]

    P. Rastogi and S. P. Mahulikar, Geometry-Based Entropy Generation Minimization in Laminar Internal Convective Micro-Flow, J. Non-Equilib. Thermodyn. 44 (2019), no. 1, 81–90.

    • Crossref
    • Export Citation
  • [27]

    P. Rastogi and S. P. Mahulikar, Theoretical Studies on Energy Degradation Estimation and Minimization in Laminar Convective Flow towards the Microscale, Heat Transf. Asian Res. 47 (2018), no. 7, 907–917.

    • Crossref
    • Export Citation
  • [28]

    P. Rastogi and S. P. Mahulikar, Optimization of Micro-Heat Sink Based on Theory of Entropy Generation in Laminar Forced Convection, Int. J. Therm. Sci. 126 (2018), 96–104.

    • Crossref
    • Export Citation
  • [29]

    A. A. Alfaryjat, A. Dobrovicescu and D. Stanciu, Influence of Heat Flux and Reynolds Number on the Entropy Generation for Different Types of Nanofluids in a Hexagon Microchannel Heat Sink, Chin. J. Chem. Eng. 27 (2019), no. 3, 501–513.

    • Crossref
    • Export Citation
  • [30]

    S. R. Hosseini and M. Sheikholeslami, Investigation of the Nanofluid Convective Flow and Entropy Generation within a Microchannel Heat Sink Involving Magnetic Field, Powder Technol. 351 (2019), 195–202.

    • Crossref
    • Export Citation
  • [31]

    P. R. Chauhan, K. Kumar, R. Kumar, M. Rahimi-Gorji and R. S. Bharj, Effect of Thermophysical Property Variation on Entropy Generation towards Micro-Scale, J. Non-Equilib. Thermodyn. 45 (2020), no. 1, 1–17.

    • Crossref
    • Export Citation
  • [32]

    B. Khlifi, B. Kalech, M. Bouterra and A. E. Cafsi, Thermodynamic Optimisation of Rectangular and Elliptical Microchannels with Nanofluids, Int. J. Exergy 25 (2018), no. 3, 252–279.

    • Crossref
    • Export Citation
  • [33]

    P. R. Chauhan, R. Kumar and R. S. Bharj, Optimization of the Circular Microchannel Heat Sink under Viscous Heating Effect Using Entropy Generation Minimization Method, Therm. Sci. Eng. Prog. 13 (2019), 100365.

  • [34]

    J. P. Holman, Heat transfer, 8th SI-metric edition, Tata McGraw-Hil Publishing Co., Ltd., New Delhi, 2003, p. 650.

  • [35]

    P. K. Mondal and S. Wongwises, Assesment of Thermodynamic Irreversibility in a Micro-Scale Viscous Dissipative Circular Couette Flow, Entropy 20 (2018), no. 1, 50.

    • Crossref
    • Export Citation
  • [36]

    Z. X. Li, F. L. Renault, A. O. C. Gómez, M. M. Sarafraz, H. Khan, M. R. Safaei, et al., Nanofluids as Secondary Fluid in the Refrigeration System: Experimental Data, Regression, ANFIS, and NN Modelling, Int. J. Heat Mass Transf. 144 (2019), 118635.

  • [37]

    N. Parlak, M. Gur, T. Engin and H. Kucuk, Exergy Analysis of Laminar Fluid Flow in Stainless Steel Microtubes, Int. J. Exergy 9 (2011), no. 4, 472–485.

    • Crossref
    • Export Citation
  • [38]

    S. Genić, B. Jaćimović and A. Petrovic, A Novel Method for Combined Entropy Generation and Economic Optimization of Counter-Current and Co-Current Heat Exchangers, Appl. Therm. Eng. 136 (2018), 327–334.

    • Crossref
    • Export Citation
  • [39]

    U. Milovancevic, B. Jacimovic, S. Genic, F. El-Sagier, M. Otovic and S. Stevanovic, Thermoeconomic Analysis of Spiral Heat Exchanger with Constant Wall Temperature, Therm. Sci. 23 (2019), no. 1, 401–410.

    • Crossref
    • Export Citation
  • [40]

    B. Jacimovic, S. Genic and D. Lelea, Calculation of the Heat Transfer Coefficient for Laminar Flow in Pipes in Practical Engineering Applications, Heat Transf. Eng. 39 (2018), no. 20, 1790–1796.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Z. Li, A. Shahsavar, K. Niazi, A. A. A. A. Al-Rashed and P. Talebizadehsardari, The Effects of Vertical and Horizontal Sources on Heat Transfer and Entropy Generation in an Inclined Triangular Enclosure Filled with Non-Newtonian Fluid and Subjected to Magnetic Field, Powder Technol. 364 (2020), 924–942.

    • Crossref
    • Export Citation
  • [2]

    D. B. Tuckerman and R. F. W. Pease, High-performance heat sinking for VLSI, IEEE Electron Device Lett. 2 (1981), no. 5, 126–129.

    • Crossref
    • Export Citation
  • [3]

    Z. X. Li, U. Khaled, A. A. Al-Rashed, M. Goodarzi, M. M. Sarafraz and R. Meer, Heat Transfer Evaluation of a Micro Heat Exchanger Cooling with Spherical Carbon-Acetone Nanofluid, Int. J. Heat Mass Transf. 149 (2020), 119124.

  • [4]

    R. Sarma, A. J. Nath, T. Konwar, P. K. Mondal and S. Wongwises, Thermo-Hydrodynamics of a Viscoelastic Fluid Under Asymmetrical Heating, Int. J. Heat Mass Transf. 125 (2018), 515–524.

    • Crossref
    • Export Citation
  • [5]

    M. Saffaripour and R. Culham, Measurement of Entropy Generation in Microscale Thermal-Fluid Systems, J. Heat Transf. 132 (2010), no. 12, 121401.

  • [6]

    L. Chen, C. Wu and F. Sun, Finite Time Thermodynamic Optimization or Entropy Generation Minimization of Energy Systems, J. Non-Equilib. Thermodyn. 24 (1999), no. 4, 327–359.

  • [7]

    Z. X. Li, A. Shahsavar, A. A. Al-Rashed, R. Kalbasi, M. Afrand and P. Talebizadehsardari, Multi-Objective Energy and Exergy Optimization of Different Configurations of Hybrid Earth-Air Heat Exchanger and Building Integrated Photovoltaic/Thermal System, Energy Convers. Manag. 195 (2019), 1098–1110.

    • Crossref
    • Export Citation
  • [8]

    M. A. Rosen, Size Considerations in Applications of Exergy Analysis, Int. J. Exergy 5 (2008), no. 3, 249–274.

    • Crossref
    • Export Citation
  • [9]

    K. Kumar, R. Kumar and R. S. Bharj, Entropy Generation Analysis due to Heat Transfer and Nanofluid Flow Through Microchannels: A Review, Int. J. Exergy 31 (2020), no. 1, 49–86.

    • Crossref
    • Export Citation
  • [10]

    T. Furukawa and W. J. Yang, Thermal Optimization of Channel Flows with Discrete Heating Sections, J. Non-Equilib. Thermodyn. 28 (2003), no. 4, 299–310.

  • [11]

    A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, 1995.

  • [12]

    H. Abbassi, Entropy Generation Analysis in a Uniformly Heated Microchannel Heat Sink, Energy 32 (2007), no. 10, 1932–1947.

    • Crossref
    • Export Citation
  • [13]

    M. Avci and O. Aydin, Second-Law Analysis of Heat and Fluid Flow in Microscale Geometries, Int. J. Exergy 4 (2007), no. 3, 286.

    • Crossref
    • Export Citation
  • [14]

    Y. M. Hung, A Comparative Study of Viscous Dissipation Effect on Entropy Generation in Single-Phase Liquid Flow in Microchannels, Int. J. Therm. Sci. 48 (2009), no. 5, 1026–1035.

    • Crossref
    • Export Citation
  • [15]

    W. A. Khan, J. R. Culham and M. M. Yovanovich, Optimization of Microchannel Heat Sinks Using Entropy Generation Minimization Method, IEEE Trans. Compon. Packag. Technol. 32 (2009), 243–251.

    • Crossref
    • Export Citation
  • [16]

    J. Guo, M. Xu, J. Cai and X. Huai, Viscous Dissipation Effect on Entropy Generation in Curved Square Microchannels, Energy 36 (2011), no. 8, 5416–5423.

    • Crossref
    • Export Citation
  • [17]

    W. H. Mah, Y. M. Hung and N. Guo, Entropy Generation of Viscous Dissipative Nanofluid Flow in Microchannels, Int. J. Heat Mass Transf. 55 (2012), no. 15-16, 4169–4182.

    • Crossref
    • Export Citation
  • [18]

    G. Ibáñez, A. López, J. Pantoja, J. Moreira and J. A. Reyes, Optimum Slip Flow Based on the Minimization of Entropy Generation in Parallel Plate Microchannels, Energy 50 (2013), 143–149.

    • Crossref
    • Export Citation
  • [19]

    G. Ibáñez, A. López, J. Pantoja and J. Moreira, Combined Effects of Uniform Heat Flux Boundary Conditions and Hydrodynamic Slip on Entropy Generation in a Microchannel, Int. J. Heat Mass Transf. 73 (2014), 201–206.

    • Crossref
    • Export Citation
  • [20]

    S. V. Prabhu and S. P. Mahulikar, Effects of Density and Thermal Conductivity Variations on Entropy Generation in Gas Micro-Flows, Int. J. Heat Mass Transf. 79 (2014), 472–485.

    • Crossref
    • Export Citation
  • [21]

    A. Ebrahimi, F. Rikhtegar, A. Sabaghan and E. Roohi, Heat Transfer and Entropy Generation in a Microchannel with Longitudinal Vortex Generators Using Nanofluids, Energy 101 (2016), 190–201.

    • Crossref
    • Export Citation
  • [22]

    L. Kuddusi, Entropy Generation in Rectangular Microchannels, Int. J. Exergy 19 (2016), no. 1, 110.

    • Crossref
    • Export Citation
  • [23]

    J. M. Cruz-Duarte, A. Garcia-Perez, I. M. Amaya-Contreras, C. R. Correa-Cely, R. J. Romero-Troncoso and J. G. Avina-Cervantes, Design of Microelectronic Cooling Systems Using a Thermodynamic Optimization Strategy Based on Cuckoo Search, IEEE Trans. Compon. Packag. Manuf. Technol. 7 (2017), no. 11, 1804–1812.

    • Crossref
    • Export Citation
  • [24]

    E. Manay, E. F. Akyürek and B. Sahin, Entropy Generation of Nanofluid Flow in a Microchannel Heat Sink, Results Phys. 9 (2018), 615–624.

    • Crossref
    • Export Citation
  • [25]

    A. A. A. A. Al-Rashed, A. Shahsavar, O. Rasooli, M. A. Moghimi, A. Karimipour and M. D. Tran, Numerical Assessment into the Hydrothermal and Entropy Generation Characteristics of Biological Water-Silver Nano-Fluid in a Wavy Walled Microchannel Heat Sink, Int. Commun. Heat Mass Transf. 104 (2019), 118–126.

    • Crossref
    • Export Citation
  • [26]

    P. Rastogi and S. P. Mahulikar, Geometry-Based Entropy Generation Minimization in Laminar Internal Convective Micro-Flow, J. Non-Equilib. Thermodyn. 44 (2019), no. 1, 81–90.

    • Crossref
    • Export Citation
  • [27]

    P. Rastogi and S. P. Mahulikar, Theoretical Studies on Energy Degradation Estimation and Minimization in Laminar Convective Flow towards the Microscale, Heat Transf. Asian Res. 47 (2018), no. 7, 907–917.

    • Crossref
    • Export Citation
  • [28]

    P. Rastogi and S. P. Mahulikar, Optimization of Micro-Heat Sink Based on Theory of Entropy Generation in Laminar Forced Convection, Int. J. Therm. Sci. 126 (2018), 96–104.

    • Crossref
    • Export Citation
  • [29]

    A. A. Alfaryjat, A. Dobrovicescu and D. Stanciu, Influence of Heat Flux and Reynolds Number on the Entropy Generation for Different Types of Nanofluids in a Hexagon Microchannel Heat Sink, Chin. J. Chem. Eng. 27 (2019), no. 3, 501–513.

    • Crossref
    • Export Citation
  • [30]

    S. R. Hosseini and M. Sheikholeslami, Investigation of the Nanofluid Convective Flow and Entropy Generation within a Microchannel Heat Sink Involving Magnetic Field, Powder Technol. 351 (2019), 195–202.

    • Crossref
    • Export Citation
  • [31]

    P. R. Chauhan, K. Kumar, R. Kumar, M. Rahimi-Gorji and R. S. Bharj, Effect of Thermophysical Property Variation on Entropy Generation towards Micro-Scale, J. Non-Equilib. Thermodyn. 45 (2020), no. 1, 1–17.

    • Crossref
    • Export Citation
  • [32]

    B. Khlifi, B. Kalech, M. Bouterra and A. E. Cafsi, Thermodynamic Optimisation of Rectangular and Elliptical Microchannels with Nanofluids, Int. J. Exergy 25 (2018), no. 3, 252–279.

    • Crossref
    • Export Citation
  • [33]

    P. R. Chauhan, R. Kumar and R. S. Bharj, Optimization of the Circular Microchannel Heat Sink under Viscous Heating Effect Using Entropy Generation Minimization Method, Therm. Sci. Eng. Prog. 13 (2019), 100365.

  • [34]

    J. P. Holman, Heat transfer, 8th SI-metric edition, Tata McGraw-Hil Publishing Co., Ltd., New Delhi, 2003, p. 650.

  • [35]

    P. K. Mondal and S. Wongwises, Assesment of Thermodynamic Irreversibility in a Micro-Scale Viscous Dissipative Circular Couette Flow, Entropy 20 (2018), no. 1, 50.

    • Crossref
    • Export Citation
  • [36]

    Z. X. Li, F. L. Renault, A. O. C. Gómez, M. M. Sarafraz, H. Khan, M. R. Safaei, et al., Nanofluids as Secondary Fluid in the Refrigeration System: Experimental Data, Regression, ANFIS, and NN Modelling, Int. J. Heat Mass Transf. 144 (2019), 118635.

  • [37]

    N. Parlak, M. Gur, T. Engin and H. Kucuk, Exergy Analysis of Laminar Fluid Flow in Stainless Steel Microtubes, Int. J. Exergy 9 (2011), no. 4, 472–485.

    • Crossref
    • Export Citation
  • [38]

    S. Genić, B. Jaćimović and A. Petrovic, A Novel Method for Combined Entropy Generation and Economic Optimization of Counter-Current and Co-Current Heat Exchangers, Appl. Therm. Eng. 136 (2018), 327–334.

    • Crossref
    • Export Citation
  • [39]

    U. Milovancevic, B. Jacimovic, S. Genic, F. El-Sagier, M. Otovic and S. Stevanovic, Thermoeconomic Analysis of Spiral Heat Exchanger with Constant Wall Temperature, Therm. Sci. 23 (2019), no. 1, 401–410.

    • Crossref
    • Export Citation
  • [40]

    B. Jacimovic, S. Genic and D. Lelea, Calculation of the Heat Transfer Coefficient for Laminar Flow in Pipes in Practical Engineering Applications, Heat Transf. Eng. 39 (2018), no. 20, 1790–1796.

    • Crossref
    • Export Citation
OPEN ACCESS

Journal + Issues

The Journal of Non-Equilibrium Thermodynamics serves as an international publication organ for new ideas, insights and results on non-equilibrium phenomena in science, engineering and related natural systems. The central aim of the journal is to provide a bridge between science and engineering and to promote scientific exchange on non-equilibrium phenomena and on analytic or numeric modeling for their interpretation.

Search

  • View in gallery

    Schematic diagram of the microchannel.

  • View in gallery

    The variation of Sgen,Fr, Sgen,HT, and Sgen,total with N corresponding to DN.

  • View in gallery

    The variation of Be with N corresponding to DN.

  • View in gallery

    The variation of DN with Be.

  • View in gallery

    The variation of the entropy generation number (NS) with N corresponding to DN.

  • View in gallery

    The variation of exergy loss (Xloss) with N corresponding to DN.

  • View in gallery

    The variation of the entropy generation number (NS), with N corresponding to DN for the numerical and explicit methods.