Circular Microchannel Heat Sink Optimization Using Entropy Generation Minimization Method

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

Schematic diagram of the microchannel.

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:

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+1−ui,jri,j+1−ri,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,j−Ti,jri+1,j−ri,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

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

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

The variation of Be with N corresponding to DN.

Figure 4

The variation of DN with Be.

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:

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

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

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

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

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]:

where B=4·qw,N″ρ·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]:

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

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

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 Sgen,Fr and Sgen,HT-rad, while Sgen,HT-ax remains almost constant for the cases under consideration.

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. The Sgen,HT decreases with increasing number of channels as a result of a decrement in the temperature gradient in the radial direction.

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. 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. 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. The EG number corresponding to N∗ is 0.1532.

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 NS-Explicit and NS-Numerical is 0.648 %.