Abstract
In this study, a critical review of thermodynamic optimum of microchannels based on entropy generation analysis is presented. Using entropy generation analysis as evaluation parameter of microchannels has been reported by many studies in the literature. In these studies, different working fluids such as nanofluids, air, water, engine oil, aniline, ethylene glycol, and non-Newtonian fluids have been used. For the case of nanofluids, “nanoparticles” has been used in various kinds such as Al2O3 and Cu, and “base fluid” has been used in various kinds such as water and ethylene glycol. Furthermore, studies on thermodynamic optimum of microchannels based on entropy generation analysis are summarized in a table. At the end, recommendations of future work for thermodynamic optimum of microchannels based on entropy generation analysis are given. As a result, this article can not only be used as the starting point for the researcher interested in entropy generation in microchannels, but it also includes recommendations for future studies on entropy generation in microchannels.
Introduction
Microscale fluid flow and heat transfer have become one of the hottest topics in mechanical engineering. At short time and length scales, interfacial effect has been increasingly important, so that the physical laws used in classical fluid flow and heat transfer are not valid anymore. For microscale, the fluid flow and heat transfer are different from macroscale ones. As current research and development on microscale fluid flow and heat transfer are reaching a plateau, it is important to make further breakthrough needed for applications of devices for microscale.
These days, microchannels can be found in many applications such as cooling of electronic devices, micro air vehicles (MAV), micro heat exchanger systems, aircraft intake de-icing, and so on. Fluid flow and heat transfer in microchannels have emerged as an important research area. This has been motivated by their different applications such as medical and biomedical use, computer chips, and chemical separations. The advent of mechanical electromechanical systems (MEMS) has opened up a new research area where noncontinuum behavior is significant. New advances in microscales are being realized and the contributions of micro heat dissipation devices are very important in this novel technology development. High heat loads of biomedical, chemical, and mechanical microsystems require heat exchangers that are very small, robust, and efficient.
The microchannels have been widely used in mechanical engineering applications. Currently, microchannels are widely used due to advancements in microscale fabricating technologies. A number of investigations have been conducted to better design the fluid flow and heat transfer in microchannel, particularly as it pertains to applications involving the thermal control of electronic devices. The analysis of entropy generation mechanism in microchannels is very important to optimize the second law performance of these energy conversion devices in microscale.
Thermodynamic optimization considers the role of entropy in engineering systems. Specifically, it addresses the issue of entropy production. Entropy production is directly related to the destruction of exergy or availability in a thermal system. Minimizing exergy destruction as means of maximizing thermodynamic performance is now at the forefront of modern thermal design techniques. In a thermodynamic optimization, entropy production or entropy production rate becomes the objective function, with constrained or unconstrained minimization in mind. This modern field is referred to as entropy generation minimization (EGM). The use of EGM allows the combined influence of thermal resistance and pressure drop to be assessed simultaneously as the heat exchanger interacts with the surrounding flow field. Past studies showed that microchannel design is dependent on its thermal resistance and pressure drop. However, EGM as a new optimization theory stated that the entropy generation rate should be also optimized. EGM is potentially a design tool to determine best geometry and operation.
In his book, Bejan 1 showed that the EGM method was dependent on the use of fluid mechanics, heat transfer, and thermodynamics in its application as shown in Figure 1. This method uses the nondimensional entropy generation number (Ns) that is defined as the entropy generation divided by the fluid heat capacity rate, which is considered the most common way of nondimensionalizing entropy generation. Its value can range between 0 and ∞.
Entropy generation minimization field covered by Bejan. 1
The difference between the exergy method and the EGM method is that exergy method uses only the first law, second law, and the properties of the environment. However, EGM characteristics are system modeling, development of the entropy generation rate as a function of the model parameters, and the ability to minimize the entropy generation rate.
Bejan 1 applied the entropy generation balance or entropy imbalance equation to a control volume of an open system. For gas–gas heat exchanger, he explained entropy generation as the sum of the entropy generation caused by finite temperature difference with frictional pressure drop
The first term on the right-hand side of equation (1) is the entropy generation rate accounting for the heat transfer irreversibility and the second term for the fluid friction irreversibility. He expressed that entropy generation (Sgen) = 0 corresponded to the highest quality, while the entropy generation (Sgen) > 0 represented poor quality.
Also, he described the relative importance of the two irreversibility mechanisms using the irreversibility distribution ratio (ϕ) that was defined as
Substituting equation (2) into equation (1) yields
Paoletti et al. 2 associated the symbol Be as an alternative irreversibility distribution parameter and defined as the ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction
Also, Benedetti and Sciubba 3 called it the Bejan number (Be). Later, Natalini and Sciubba 4 also introduced Bejan number (Be) using equation (4).
It is clear from equation (4) that Be = 1 that occurs at ϕ = 0 corresponds to the case at which the irreversibility is dominated by the heat transfer effects. However, Be = 0 corresponds to the case at which the irreversibility is dominated by the fluid friction effects. Also, Be = 0.5 that occurs at ϕ = 1 corresponds to the case at which the heat transfer irreversibility and the fluid friction irreversibility are equal. In addition, Be = (1 + 50.5)/2 ≅ 0.618 that occurs at ϕ≅ 0.618 corresponds to the case at which the Bejan number (Be) and the irreversibility distribution ratio (ϕ) are equal.
It should be noted that the Be definition in equation (4) should not be confused with other Bejan number (Be) used in fluid mechanics, 5 heat transfer, 6 and mass transfer. 7 Recently, Awad and Lage 8 extended the Bejan number (Be) to a general form that can be used in fluid mechanics, heat transfer, and mass transfer.
EGM method has been used for the thermodynamic optimization in many applications. For example, Awad and Muzychka 9 wrote recently a book chapter on thermodynamic optimization of heat exchangers. In this article, application of EGM in microchannel design is reviewed and discussed in detail.
This article consists of four sections: “Introduction,”“Literature review,”“Recommendations for future studies,” and “Summary and Conclusions.” The author used the same style in writing his recent article about condensation heat transfer in microchannels and minichannels. 10
Literature review
In the literature, there are some recent review articles that include few studies on entropy generation in microchannels. For example, Mahian et al. 11 presented recently a review of entropy generation in nanofluid flow. Their review includes four studies on entropy generation in microchannels using nanofluids, that is, dilute suspensions of nanoparticles in liquids, as the working fluids that may exhibit remarkable heat transfer characteristics, especially for heat removal in microdevices. These four studies were done by Singh et al., 12 Li and Kleinstreuer, 13 Mah et al., 14 and Shalchi-Tabrizi and Seyf. 15
This article includes many studies on entropy generation in microchannels. The author of this article on entropy generation in microchannels will cover different working fluids such as nanofluids, air, water, engine oil, aniline, ethylene glycol (EG), and non-Newtonian fluids. This article is classified into two categories. The first category is related to studies on entropy generation in microchannels without using nanofluids as the working fluids. The second category is related to studies on entropy generation in microchannels using nanofluids as the working fluids.
Entropy generation in microchannels in the absence of nanofluids
Chen 16 calculated and analyzed entropy generation and transfer in microchannel flows for different thermal boundary conditions. Due to the small flow cross-sectional area, the researcher neglected the fluid temperature variation in the lateral direction and developed and used a laterally lumped model in the first and second law analyses. He took into consideration heat conduction in the flow direction because the Péclet numbers (Pe) of microchannel flows were typically low. Computed fluid temperature and entropy generation rate were cast into dimensionless forms; thus, it could be applied to various fluids and channels of various sizes and configurations. He found that local entropy generation rate was only dependent upon the temperature gradient in the flow direction. His optimization results of microchannel flows exchanging heat with their surroundings indicated that the optimal fluid temperature distribution was a linear one.
Abbassi 17 investigated the entropy generation in a uniformly heated microchannel heat sink (MCHS). Figure 2 shows the schematic diagram of MCHS. He used analytical approach to solve forced convection problem across MCHS. This analytical approach was a porous medium model based on extended Darcy equation for fluid flow and two-equation model for heat transfer. Simultaneously, closed-form velocity solution in a rectangular channel was employed to capture z-directional viscous effect diffusion and its pronounced influence on entropy generation through fluid flow. Subsequently, governing equations were cast into dimensionless form and solved analytically. Then, second law analysis of problem was conducted on the basis of obtained velocity and temperature fields, and expressions for local and average entropy generation rate were derived in a dimensionless form. Then, average entropy generation rate was utilized as a criterion for assessing the system performance. At the end, the effect of influential parameters such as channel aspect ratio (AR, αS), group parameter (Br/Ω), thermal conductivity ratio (C), and porosity (ε) on thermal and total entropy generation was investigated. In order to examine the accuracy of the analysis, the results of thermal evaluation were compared to one of the previous investigations conducted for thermal optimization of MCHS.
Schematic diagram of microchannel heat sink (MCHS).
Hung 18 studied analytically the viscous dissipation effect on entropy generation in fully developed forced convection for single-phase non-Newtonian fluid flow in circular microchannels. The researcher obtained closed-form solutions of the temperature distributions in the radial direction for the models with and without viscous dissipation term in the energy equation in the first law analysis. He compared the two models by analyzing their relative deviations in dimensionless entropy generation and Bejan number (Be) for various Brinkman number (Br) and power law index (n) in the second law analysis. Under certain conditions, he found that the viscous dissipation effect on entropy generation in microchannels was significant and should not be neglected.
Yazdi et al. 19 presented an analysis of entropy generation of forced convection heat transfer of liquid fluid over the horizontal surface with embedded open parallel microchannels at constant heat flux boundary conditions. The researchers transformed the governing partial differential equations subjected to slip flow boundary conditions into a set of ordinary differential equations using similarity variables. They solved numerically these ordinary differential equations using a fourth-order Runge–Kutta and shooting method. They considered entropy production related to embedded open parallel microchannels. They formulated the entropy generation rate by an integral of local entropy generation subjected to constant heat flux boundary condition. They investigated and discussed in detail the entropy generation rate for different values of slip coefficient. They showed that the embedded open parallel microchannels within the surface could sufficiently reduce both friction and thermal irreversibilities of liquid fluid through slip flow conditions.
Hung 20 studied the viscous dissipation effect on entropy generation in fully developed forced convection for single-phase liquid flow in a circular microchannel under imposed uniform wall heat flux. The researcher obtained closed-form solutions of the radial temperature profiles for the models with and without viscous dissipation term in the energy equation in the first law analysis. He investigated the variations of dimensionless entropy generation and Bejan number (Be) as a function of the radial distance for various Brinkman number (Br) and dimensionless heat flux in the second law analysis. He compared the two models by analyzing their relative deviations in dimensionless entropy generation and Bejan number. Also, he performed comparisons for average dimensionless entropy generation and average Bejan number. He analyzed and discussed contribution of heat transfer irreversibility and fluid friction irreversibility to the deviations. Under certain conditions, he found that the viscous dissipation effect on entropy generation in microchannel was significant and should not be neglected.
Khan et al. 21 employed an EGM procedure to optimize the overall performance of MCHSs. The researchers developed new general expressions for the entropy generation rate by considering an appropriate control volume and applying mass, energy, and entropy balances. They investigated the influence of channel AR, fin spacing ratio, heat sink material, Knudsen numbers, and accommodation coefficients on the entropy generation rate in the slip flow region. They used analytical or empirical correlations for heat transfer and friction coefficients, where the characteristic length was used as the hydraulic diameter of the channel. In addition, a parametric study was performed to show the effects of various design variables on the overall performance of MCHSs.
Sadeghi et al. 22 carried out the second law of thermodynamics analysis for steady-state hydrodynamically and thermally fully developed laminar gas flow in annulus microchannels with asymmetrically heated walls. The researchers took into consideration the rarefaction effects using first-order slip velocity and temperature jump boundary conditions. Also, they included viscous heating for both the hot wall and the cold wall cases. Using the velocity distribution obtained in earlier works, they solved the energy equation to get analytically the temperature distribution and consequently to compute the entropy generation rate. They discussed the effects of rarefaction and the annulus geometrical AR on velocity distribution. They showed in graphical form and also discussed in detail the complicated interactive effects of rarefaction, viscous dissipation, the ratio of Brinkman number to dimensionless temperature difference, annulus geometrical AR, and asymmetry on entropy generation rate and Bejan number. They compared the analytical results obtained with those available in the literature and observed an excellent agreement. They found that the effect of the wall heat flux ratio on entropy generation was negligible at great values of the ratio of Brinkman number to dimensionless temperature difference (the group parameter, Br/Ω), while the effect of increasing values of the annulus geometrical AR was to severely increase entropy generation. The entropy generation decreased as Knudsen number increased; however, the effect of increasing values of Brinkman number and the ratio of Brinkman number to dimensionless temperature difference was to increase entropy generation.
Jafari and Ghazali 23 derived an optimization procedure using EGM method with the entropy generation rate for a circular MCHS based upon thermal resistance and pressure drop. The researchers solved the equations using MATLAB and compared the obtained results to similar past studies. They investigated the influences of channel diameter, number of channels, heat flux, and pumping power on the entropy generation rate and Reynolds number. They utilized analytical correlations for heat transfer and friction coefficients. They observed a minimum entropy generation for a total number of microchannels (N) = 40 and channel diameter of 90 μm. They concluded that the circular MCHS was on its optimum operating point for N = 40 and channel hydraulic diameter of 90 μm, based on second law of thermodynamics.
Jung and Kim 24 examined analytical solutions for entropy generation rate distribution associated with heat transfer and fluid friction in MCHSs. The researchers modeled MCHSs as a porous medium through which fluid flowed. They obtained analytical solutions using velocity and temperature distributions of MCHSs that were based on the modified Darcy model for fluid flow and the two-equation model for heat transfer. Using the analytical solution, they obtained the entropy generation of heat sinks. They studied the effects of height, channel width, and fin thickness on the entropy generation rate and performed thermal optimization of heat sink.
Zhao and Liu 25 analyzed the entropy generation of electro-osmotic flow in two-dimensional open-end and closed-end microchannels. The microchannels consisted of two parallel plates, which were made of silicon glass exhibiting electro-osmotic influences and separated by a distance (H), and the length of the microchannel was L. The microchannel was filled with an ionized solution. The two closed-end walls were modeled as electrodes in the closed-end microchannel, and it was assumed that there was no electric double layer (EDL) formation on the electrodes at the closed-end wall. The researchers used a rigorous mathematical model for describing electro-osmotic flow in their study. They simulated numerically the entropy generations of electro-osmotic flow due to heat conduction, viscous dissipation, and Joule heating. They found that the volumetric entropy generation rates due to heat conduction and viscous dissipation were maximum near the microchannel wall, and the volumetric entropy generation rate due to Joule heating was maximum at the center of the microchannel. Because of the Joule heating effect, the heat conduction entropy generation number and Joule heating entropy generation number increased with the applied electric field (E), and the entropy generation of viscous dissipation could be neglected in the open-end and closed-end electro-osmotic flow. When the temperature increment due to Joule heating was larger than the temperature difference between the inlet and the top wall, the electro-osmotic flow entropy generation due to Joule heating would take the major percent in the total entropy generation. When the temperature difference between the inlet and the top wall was larger than the temperature increment due to Joule heating, the electro-osmotic flow entropy generation due to heat conduction would take the major percent in the total entropy generation.
Galvis and Culham 26 used the EGM method to find the optimum channel dimensions in micro heat exchangers with a uniform heat flux. With this approach, the researchers considered simultaneously pressure drop and heat transfer in the microchannels during the optimization analysis. They developed a computational model to find the optimum channel depth knowing other channel geometry dimensions and coolant inlet properties. They assumed that the flow was laminar and both hydrodynamically and thermally fully developed and incompressible. However, they introduced the Hagenbach’s factor for rectangular channels obtained by Steinke and Kandlikar 27 to take into account the developing length effect in the friction losses. The microchannels were assumed to have an isoflux or isothermal boundary condition, no-slip flow, and fluid properties that had dependency on temperature accordingly. For these particular case studies, the pressure drop and heat transfer coefficient for the isoflux boundary condition were higher than the isothermal case. When the channel size decreased, they found higher heat transfer coefficient and pressure drop. The optimum channel geometry that minimized the entropy generation rate tended to be a deep, narrow channel.
Sadeghi et al. 28 investigated analytically the entropy generation in laminar forced convection of a Newtonian fluid through a slit microchannel by taking the viscous dissipation effect, the slip velocity, and the temperature jump at the wall into account. The researchers considered flow to be hydrodynamically fully developed but thermally developing. They solved the energy equation by means of integral transform. Their results demonstrated that to increase Knudsen number (Kn) was to decrease entropy generation, while the influence of increasing values of Brinkman number (Br) and the group parameter (Br/Ω) was to increase entropy generation. Also, they disclosed that the average entropy generation number over the cross section of channel in the thermal entrance region was an increasing function of axial coordinate.
Ibáñez and Cuevas 29 studied the EGM in a microchannel flow subjected to electromagnetic interactions, as occurred in a magnetohydrodynamic (MHD) micropump. The researchers used the entropy generation rate as a tool to assess the intrinsic irreversibilities present in the microchannel owing to viscous friction, heat flow, and electric conduction. They considered the flow in a parallel plate microchannel produced by a Lorentz force created by a transverse magnetic field and an injected electric current. They assumed a thermally fully developed flow and conducting walls of finite thickness. They solved analytically the conjugate heat transfer problem in the fluid and solid walls using thermal boundary conditions of the third kind at the outer surfaces of the walls and continuity of temperature and heat flux across the fluid–wall interfaces. They used velocity, temperature, and current density fields in the fluid and walls to calculate the global entropy generation rate. They determined conditions under which this quantity is minimum for specific values of the geometrical and physical parameters of the system. Also, they calculated and explored the Nusselt number for various conditions. Their results could be used to determine optimized conditions that led to a minimum dissipation consistent with the physical constraints demanded by the microdevice.
Saffaripour and Culham 30 presented a new nonintrusive and whole field method for the measurement of entropy generation in microscale thermal–fluid devices. Their method provided the entropy generation distribution in the device, thus enabling the designers to find and modify the areas producing high energy losses characterized by large entropy production rates. The researchers obtained the entropy generation map by post-processing the velocity and temperature distribution data, measured by micro particle image velocimetry and laser-induced fluorescence methods, respectively. The velocity and temperature measurements led to the frictional and thermal terms of entropy generation. One main application of their method was optimizing the efficiency of MCHSs, used in cooling of electronic devices. The minimum amount of entropy generation determined the optimum design parameters of heat sinks, leading to highest heat removal rates and, at the same time, the lowest pressure drop across the heat sink. To show the capability of their technique, the entropy generation field in the transition region between a 100-μm-wide and a 200-μm-wide rectangular microchannel was measured. They used their method to measure thermal and frictional entropy generation rates in three different flow area transition geometries. They selected three geometries for the transition at the entrance and exit of the narrow channel in their study, namely, rectangular, triangular, and circular. The narrow channel was 100 μm wide and 20 mm long, located between 200-μm-wide and 10-mm-long channels to ensure parallel and disturbance free flow at the entrance to the narrow channel. The microchannels depth was 100 μm everywhere. Their results could be used to determine which geometry had the highest thermal and hydraulic efficiencies. Their measurement results indicated that the frictional and thermal entropy generation rates had inverse trends of variation with flow rate. Thus, there was an optimum flow rate at which the losses due to the combined frictional and heat transfer influences were minimized. The thermal entropy generation rate was orders of magnitude larger than the frictional entropy generation rate. Practical limitations did not allow the increase in the flow rate to capture the optimum point.
Sadeghi and Saidi 31 carried out the second law of thermodynamics analysis for steady-state fully developed laminar gas flow in a parallel plate microchannel with asymmetrically heated walls. The rarefaction effects as well as viscous heating effects are taken into consideration. Closed-form expressions are obtained for velocity and temperature distributions and entropy generation rates. The results demonstrate that increasing values of the wall heat flux ratio result in greater entropy generation for positive Brinkman numbers, whereas the opposite is true for negative values of Brinkman. However, the effect of the wall heat flux ratio on entropy generation becomes insignificant at high values of the group parameter (Br/Ω) and Péclet number (Pe). The entropy generation decreases as Knudsen (Kn) and Péclet (Pe) numbers increase; however, the effect of increasing values of Brinkman number (Br) and the group parameter (Br/Ω) is to increase entropy generation. Also, it is realized that the influences of rarefaction on entropy generation are negligible for low Péclet number flows.
Al-Obaidi 32 used second law analysis for a steady-state cross-flow microchannel heat exchanger (MCHX) because this type of heat exchangers was known for its higher heat transfer coefficient and higher area per volume ratio. As a result, broad range studies were being carried out to optimize its performance and minimize its inefficiencies. The researcher employed entropy generation and exergy loss to investigate a multiport serpentine slab MCHX with EG-water and air as the working fluids. She used conservation of energy and the increase in entropy principles to create a mathematical model that used various parameters such as heat capacity rate ratio, fluids inlet temperatures, effectiveness, and pressure drop for obtaining entropy generation. Results were found on the basis of the behavior of the entropy generation number (Ns) with the key parameters. She found a good agreement between the predicted and the measured results.
Kuddusi 33 studied fully developed gaseous slip flow in trapezoidal silicon microchannels. The researcher obtained friction factor, Nusselt number, and entropy generation in the microchannel. He explored the rarefaction effect, AR, and viscous dissipation and specified the range of Brinkman number (Br) in which viscous dissipation influence was important and could not be neglected. He applied the continuum approach with the velocity slip and temperature jump condition at the solid walls to develop the mathematical model of problem in the trapezoidal microchannel. Transformation of trapezoidal geometry to a square provided the ease of application of finite difference method in the mathematical model solution. The viscous dissipation effect was quantified by Brinkman number. The calculated Brinkman number for common engineering applications even with limiting operational and geometric conditions was found to be <0.005. He observed that viscous effect for applications with Br < 0.005 could be neglected. The region in which viscous dissipation effect could not be neglected was specified as Br > 0.005. Decreasing influence of rarefaction and increasing influence of Brinkman number on irreversibility due to all sources, excluding axial conduction, were established. He specified the dominant source of irreversibility in total irreversibility as a function of Brinkman number (Br).
Nouri-Borujerdi 34 used the second law of thermodynamics to optimize microchannels sizes in two-phase flows and considered the overall performance of microchannels. His numerical solution of the conservation equations provided pressure drop and heat transfer rate. There was an AR that generated the lowest entropy generation rate. His results indicated that decreasing the AR contributed to the entropy generation reduction by heat transfer, because smaller channel width allowed for less heat transfer rate. In contrast, decreasing the AR contributed the increasing of pressure drop and more entropy generation was produced. Also, his results showed that the lowest entropy generation moved toward the higher AR when the wall heat flux or vapor quality increased.
Tu et al. 35 studied fluid flow and entropy generation characteristics of the channel geometry in X-shaped microchannels. The researchers investigated the channel geometry influence on the mixing performances in the X-shaped microchannels. They considered seven various tested channels consisting of shrunk channel, normal channel, and magnified channel, made of acrylic fabric with the width ranging from 0.7 to 1.3 mm, and used water as the working fluid. Water was injected to microchannel at various mass flow rates, over a wide range of flow conditions, 0.52 < Re < 718. They deliberated numerical simulation of the entropy generation, temperature gradient, velocity vector, and pressure drop with experiment. Through the evaluations of the overall entropy generation in the whole flow domain, they found that magnified channels had the lower entropy generation and best mixture of performance for Re < 136.68. For Re > 136.68, the transition formed early from laminar flow in the smallest the channel geometry. The unsteady flow was an advantage for mixing in the limited mixing area. As a result, they generated the best mixing performance.
Guo et al. 36 investigated numerically the viscous dissipation influence on the thermodynamic performance of the curved square microchannels in laminar flow. The researchers adopted the classical Navier–Stokes equations and selected aniline and EG as the working fluids. Their results showed that the heat transfer entropy generation number and frictional entropy generation number augmented relatively under viscous dissipation influence for the case of fluid heated, and the opposite results could be found for the case of fluid cooled. The heat transfer entropy generation number increased with Reynolds number at large Reynolds number region under viscous dissipation influence when EG was heated. The total entropy generation number extremum existed for aniline, and the extremum occurred earlier when aniline was heated than when aniline was cooled. The smaller the curvature radius was, the earlier the extremum appeared. The extremum did not occur for EG due to the predomination of frictional entropy generation in the total entropy generation.
Odukoya et al. 37 conducted a transient heat transfer and entropy analysis to investigate the processes of thermocapillary droplet motion in closed microchannels. The researchers presented both theoretical predictions and experimental data for time-dependent temperature changes during the droplet acceleration. They developed a predictive model of the entropy production due to thermal and fluid irreversibilities in the microchannel. They modeled thermocapillary, pressure and friction forces within the droplet, as well as surface tension hysteresis during start-up of the droplet motion. They measured experimentally the spatial temperature change in the axial direction, as well as the displacement of the droplet over time. They reported the variation of the entropy generation number for open and closed channels. They obtained close agreement between the predicted and experimental data. Their results showed that water droplets had lower thermal and fluid irreversibilities than toluene and mineral oil.
Yazdi et al. 38 presented a new design of open parallel microchannels embedded within a permeable continuous moving surface due to reduction of exergy losses in MHD flow at a prescribed surface temperature (PST). The researchers formulated the entropy generation number by an integral of the local rate of entropy generation along the surface width based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. They substituted the velocity, the temperature, the velocity gradient, and the temperature gradient adjacent to the wall into the equation resulting from the momentum and energy equations obtained numerically by an explicit Runge–Kutta (4, 5) formula, the Dormand–Prince pair, and shooting method. Also, they presented and discussed in detail the entropy generation number (Ns), as well as the Bejan number (Be), for different values of the involved parameters of the problem.
Alquaity et al. 39 studied laminar flow in a microchannel in the presence of phase change material particles and examined entropy generation rate in the channel. The researchers investigated the effect of various parameters such as mass flow rate, heat flux, and particle volume concentrations on the volumetric entropy generation rate due to heat transfer and fluid friction. They predicted the flow and the temperature fields using a numerical method incorporating the fluid with effective thermophysical properties in the simulations. The entropy generation rate due to heat transfer was the maximum at high volume concentrations, mass flow rates, and heat flux. Entropy generation rate due to fluid friction was the maximum at high mass flow rates and volume concentrations of particles and at low heat flux. Entropy generation rate due to heat transfer was dominant at low mass flow rates; however, at high mass flow rates, entropy generation rate due to fluid friction became comparable to entropy generation rate due to heat transfer.
Guo et al. 40 investigated numerically the influence of temperature-dependent viscosity on the thermodynamic performance of the curved square microchannel in laminar flow in terms of entropy generation. The researchers adopted the classical Navier–Stokes equations and constant wall temperature boundary conditions and selected aniline and EG as the working fluids. Their results showed that the Nusselt number, heat transfer entropy generation number, and frictional entropy generation number were higher for the temperature-dependent viscosity than for the constant viscosity when aniline was cooled. However, the opposite conclusions could be drawn when aniline was heated. The total entropy generation number extrema existed for the cases of aniline cooled and heated. The differences between the results obtained with and without considering temperature-dependent viscosity were more obvious when aniline was cooled than when aniline was heated. The difference between the Brinkman numbers (Br) obtained with and without considering temperature-dependent viscosity grew as the mass flow rate increased when EG was heated. The temperature-dependent influence on entropy generation was more pronounced for EG than for aniline, since the former had larger viscosity than the latter.
Askar et al. 41 used second law analysis for a steady-state multiport serpentine slab cross-flow MCHX to analyze its thermodynamic performance. MCHX was well known for its higher heat transfer coefficient and higher area per volume ratio. The researchers used conservation of energy and the increase in entropy principles to create a mathematical model. They used various parameters in their mathematical model such as heat capacity rate ratio, fluids inlet temperatures ratio, effectiveness, and pressure drop to obtain the entropy generation. They obtained results on the basis of the behavior of the dimensionless entropy generation number with the key parameters. They found a good agreement between the predicted and the measured results.
Yazdi et al. 42 investigated the entropy generation in a laminar, electrically conducting fluid flow past open parallel microchannels embedded in a horizontal stationary surface subject to a transverse magnetic field at PST. The researchers formulated the entropy generation number by an integral of the local rate of entropy generation along the surface width based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. They substituted the velocity, the temperature, the velocity gradient, and the temperature gradient adjacent to the surface wall into the equation resulting from the momentum and energy equations obtained numerically by the Dormand–Prince pair and shooting method. They calculated, depicted graphically, and discussed in detail the entropy generation number, as well as the Bejan number (Be). They concluded that an increase in the open microchannels width tended to decrease entropy generation number along the surface.
Escandón et al. 43 analyzed the entropy generation rate in a parallel flat plate microchannel under mixed electro-osmotic and pressure-driven flow with a non-Newtonian fluid. The researchers considered a fully developed flow, and the fluid obeyed a constitutive relation based on a simplified Phan-Thien–Tanner model. They solved asymptotically the temperature distributions of a conjugate heat transfer problem in the microchannel in steady state and then obtained expressions for dimensionless local and average entropy generation rate. They showed the effect of dimensionless parameters involved on the entropy generation rate: the viscoelastic parameter, the ratio of pressure forces to electro-osmotic forces, the Péclet number (Pe), the normalized power generation term, the dimensionless temperature difference, the ratio of microchannel thickness to length, the ratio of microchannel wall thickness to length, and the conjugation parameter. This parameter set controlled directly the thermal performance of this microchannel. They predicted that the entropy generation rate was basically dominated by the Joule heating effect.
Ibáñez et al. 44 studied analytically the influences of slip flow on heat transfer and entropy generation by considering the conjugate heat transfer problem in microchannels. The researchers solved analytically the heat transfer equations in the fluid and the finite thickness walls of the microchannel using thermal boundary conditions of the third kind at the outer surfaces of the walls and continuity of temperature and heat flux across the fluid–wall interfaces. After obtaining the analytic solutions for the velocity and temperature fields in the fluid and walls of microchannel, they discussed in detail and investigated the entropy generation rate considering slip flow and convective effects, simultaneously. Their results showed that the global entropy generation rate was minimized for certain suitable combination of the geometrical and physical parameters of the system. It was possible to find an optimum slip velocity that led to a minimum global entropy generation rate. Also, they calculated and explored the Nusselt number for various conditions. They derived an optimum value of the slip length that maximized the heat transfer.
Bermejo et al. 45 presented an approach for optimizing the design of the microchannel evaporator for space electronics cooling based on the EGM because the increasing heat dissipation from electronic devices on board satellites made it necessary to find solutions for their cooling. In their study, 20 electronic components in series needed to dissipate a heat flux of 20 kW/m2 owing to microevaporators mounted in a refrigeration system. The researchers combined a steady-state three-dimensional conduction model with thermohydraulic flow boiling models valid for microchannels to solve this thermal problem. In order to calculate the two-phase heat transfer coefficient, three flow boiling correlations proposed in the literature and developed for microchannels were used: correlations of Lazarek and Black, 46 Thome et al., 47 and Kandlikar and Balasubramanian. 48 In order to calculate the two-phase pressure drop, two pressure drops were used: the homogeneous model 49 and the Müller-Steinhagen and Heck 50 model. They found that the Lazarek and Black 46 correlation led to the lowest heat transfer coefficient, and the homogeneous model 49 led to the highest pressure drop along the 20 equipments. This case represented the most unfavorable case for the design. Therefore, it was retained for the optimization process. They found that the best design corresponded to an AR (ratio between height and width) of around 8.8. Also, they analyzed the sensitivity of their results to the choice of the flow boiling models.
Escandón et al. 51 studied the entropy generation rate in a purely electro-osmotic flow of a non-Newtonian fluid in a parallel flat plate microchannel. The researchers used the power law model for the rheological constitutive equation of the fluid under consideration. They obtained the entropy generation rate as an asymptotic solution to the conjugate heat transfer problem between the fluid and the solid walls of the microchannel. They predicted the effect of the following dimensionless parameters on the entropy generation rate: the flow behavior index (n), the electrokinetic parameter (κ), the well-known Péclet number (Pe), the normalized power generation term (Λ), the dimensionless temperature difference (Ω), the ratio of the microchannel thickness to the microchannel length (AR), the ratio of the microchannel wall thickness to the microchannel wall length (ε), and a conjugate heat transfer parameter (α), which related the competition between the conductive heat in the microchannel wall and the conductive heat in the laminar flow. This set of parameters directly determined the thermal performance of the microchannel model. They predicted that the entropy generation was dominated by Joule heating, and the fluid friction contribution was negligible.
Matin and Khan 52 studied the entropy generation analysis of heat and mass transfer in a slit microchannel. The researchers assumed that the flow to be due to a combination of the electro-osmotic body force and the pressure gradient. They considered a fully developed flow with uniform flux at the surface in their analysis. They obtained analytical solutions for velocity, temperature, and concentration fields. Then, they utilized the gradients of these fields in the entropy generation analysis. They presented graphically and discussed in detail the results.
Yazdi et al. 53 examined embedded open parallel microchannels within a micropatterned permeable surface for reducing entropy generation in MHD fluid flow in microscale systems. The researchers obtained a local similarity solution for the transformed governing equations. First, they casted the governing partial differential equations along with the boundary conditions into a dimensionless form. Then, they solved numerically the reduced ordinary differential equations via the Dormand–Prince pair and shooting method. They formulated the dimensionless entropy generation number by an integral of the local rate of entropy generation along the surface width based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. At the end, they investigated the entropy generation numbers, as well as the Bejan number (Be). They found that surface-embedded microchannels could successfully reduce entropy generation in the presence of an applied magnetic field.
Zhai et al. 54 investigated numerically the performance analysis of three types of microchannels with fan-shaped reentrant cavities and different ribs based on field synergy principle and entropy generation analysis. These three types of microchannels were the microchannel F-C (with circular ribs), the microchannel F-R (with rectangular ribs), and the microchannel F-Trp (with trapezoidal ribs). The researchers compared the results obtained with the open literature. Their results showed that the influence of various types of ribs on the overall performance was evident. The thermal enhancement factor (η) of the microchannel F-R (with rectangular ribs) was always the lowest, depending on the range of Reynolds number. Also, they proposed various comprehensive performance criteria, based on the first and second law of thermodynamics, to assess the relative merit of every microchannel. They found that the microchannel F-Trp (with trapezoidal ribs) was the most promising configuration when Re < 300, while the microchannel F-C (with circular ribs) had the best performance when Re > 300. According to field synergy principle, the heat transfer enhancement could be attributed to the good synergy between velocity vector and temperature gradient. In addition, thermal boundary was disturbed after adding ribs and temperature difference between the channel wall and fluid was small, thus the corresponding irreversibility (entropy generation) reduced and heat transfer rate improved.
Elazhary and Soliman 55 investigated entropy generation during laminar fully developed flow in parallel plate microchannels with an imposed pressure gradient and a uniform wall heat flux taking into consideration the effects of an EDL at the solid–fluid interfaces and viscous dissipation due to internal fluid friction. The researchers calculated the rates of local volumetric and total entropy generation per unit channel volume due to heat transfer and viscous dissipation using analytical expressions for the velocity and temperature profiles. Comparing results from expressions using the local-temperature formulation to others using a reference-temperature formulation showed that the latter was inaccurate for low Brinkman number or small channel sizes. They presented and discussed influences of Brinkman number (Br), Péclet number (Pe), channel size, and zeta-potential on the rate of entropy production. Entropy production due to heat transfer was a maximum, while entropy production due to viscous dissipation and the ratio of pumping power to heat transfer was a minimum within a specific range of zeta-potential. These trends were attributed to the influences of EDL on the velocity and temperature profiles.
Ibáñez et al. 56 studied analytically the influences of wall heat flux boundary conditions, wall to fluid thermal conductivity ratio, and slip flow on heat transfer and entropy generation by considering the conjugate heat transfer problem in microchannels. The researchers solved analytically the heat transfer equations in the fluid and the finite thickness walls of the microchannel using uniform heat flux boundary conditions at the outer surfaces of the walls with appropriate continuity of temperature and heat flux at the fluid–wall interfaces. They utilized exact analytic solutions for the velocity and temperature fields in the fluid and walls of microchannel to compute the entropy generation rate. They integrated the latter in the whole region of analysis so that the finite dimensions of the device were considered to get the global entropy generation rate. At the end, they discussed in detail and investigated the entropy generation considering combined influences of wall and hydrodynamic slip. They found optimum values of heat flux across the microchannel walls where the global entropy generation was minimum. They gave special attention to the wall heat flux influence on optimal values of other parameters. The optimum values of both the slip length and wall to fluid thermal conductivity ratio, where the entropy generation was minimum, decreased with the wall heat flux. In addition, they found optimum values of Péclet number with minimal entropy for certain suitable combination of geometrical and physical parameters of the system.
Anand 57 studied heat transfer and entropy generation of pressure-driven flow of a power law fluid in a microchannel subjected to uniform heat flux boundary condition at the walls. The researcher modeled the slip at the walls through three various slip laws, namely, nonlinear Navier slip law, Hatzikiriakos slip law, and asymptotic slip law. He solved analytically all the governing equations. He showed the influence of different friction coefficients of the slip laws on velocity distribution, temperature distribution, Nusselt number, entropy generation rate, and Bejan number. He discussed in detail the reason and the justification behind the trends observed. He made a comparison between Hatzikiriakos slip law and asymptotic slip law with regard to Nusselt number and average entropy generation rate. For the same slip coefficients, he observed that the value of Nusselt number as predicted by Hatzikiriakos slip law was higher, and average entropy generation rate was lower than the corresponding values predicted by asymptotic slip law; and this difference increased with the increase in the value of slip coefficients.
Prabhu and Mahulikar 58 investigated the effects of density and thermal conductivity variations on entropy generation in gas micro-flows because high-temperature gradients in heat sinks of microdevices caused substantial property variations in gas micro-convection that constituted an important strategic research area in nonrarefaction scaling influences. The researchers solved numerically compressible Navier–Stokes equations incorporating temperature-dependent density and thermal conductivity variations, for steady, viscous flow of gas at subsonic speeds (Mach number (Ma) << 1). They chose circular micro-pipe geometry, subjected to constant wall heat flux. Their key observations were: In addition to known influences, two additional physical mechanisms increasingly surface at microscale; and also determined the micro-convection characteristics within continuum regime. They were induced radial convection due to density variations in the radial and axial directions and induced axial conduction due to thermal conductivity variations along the flow. High density variations in the radial and axial directions caused velocity gradients and increased fluid friction irreversibility. High thermal conductivity variations caused flattening of temperature profile, induced axial conduction, and substantially increased entropy generation. Their results highlighted the need for incorporating fluid property variation in thermal designs of heat sinks for microdevices. Two-way linked between velocity profile and temperature, effect entropy generation. Corner influences at the entrance and exit effect entropy generation in all cases were studied.
Aghanajafi et al. 59 evaluated the radiation and dissipation influences in the entropy generation in microtube. The researchers solved governing equations using analytical solution. Fluid flow in the entrance area was laminar and independent of time, and radiation was also a fluid property. Results in their work were extracted with dimensionless numbers Knudsen (Kn), Brinkman (Br), and Prandtl (Pr) and compared with the case without radiation. They found that entropy generation decreased with increasing Knudsen number (Kn). They observed that increasing Brinkman number (Br) augmented the entropy generation.
Shi and Dong 60 presented entropy generation rate accounting for heat transfer and flow friction for the flow in microchannel with staggered pin fin arrays, clearance existing between the pin fin tip and the shroud plate. Figure 3(a) and (b) shows the schematic representation of the pin fin arrays with tip clearance in a microchannel flow. Within the scope of their work, entropy generation rate due to heat transfer was levels of magnitude higher than that from flow friction that was nonetheless not negligible considering its scaling influence on the pumping power consumption for all cases under investigation. For the pin fin structure with high AR (height to diameter), the tip clearance influence was found more pronounced with a conductive nature competing with the convective. When the AR was low, the convective influence dominated the entropy generation variation by heat transfer and flow friction, and the clearance gap influence was favored by higher AR. The researchers applied the entropy minimization method to seek for an optimal combination of all impact factors under investigation. They obtained the Pareto frontier along with its corresponding solution sets using Multiobjective Optimization Genetic Algorithm. The solution sets acquired for the scenario with high-aspect-ratio pin fin fell within the region of lower pin fin density where the trade-off between the convective and the conductive influences was identified. However, dominated by the convective nature, the solution sets for cases with lower AR were in principle located at the upper bounds.
Schematic representation of the pin fin arrays with tip clearance in a microchannel flow: (a) top view and (b) side view.
Jiang et al. 61 investigated numerically the volumetric entropy generation rate distributions and exergy losses of fuel lean premixed CO/H2/air flames in a microscale cylindrical channel at atmospheric pressure. Their numerical simulation predicted the gas properties such as temperature, density, and species concentrations by the detailed gas phase chemistry and transport. The researchers calculated the entropy generation rates due to the effects of chemical reaction, thermal conduction, and mass diffusion based on the gas properties. They performed a detailed parametric study of the effect of CO/H2/air flow velocity and CO mass fraction on the volumetric entropy generation rate. The entropy generation rate induced by chemical reaction was suppressed by adding more CO to the mixture. However, higher thermal conduction–induced entropy generation was inspected in the mixture with more CO. The total entropy generation rate increased with the inlet velocity. This was the first systematical analysis of the volumetric entropy generation rate distributions of CO/H2/air flames in the microchannel. Their analysis could be applied to other power conversion systems that utilized CO/H2/air mixture.
Mohammadi and Moghadam 62 examined the fully developed slip flow of non-Newtonian Bingham plastic fluids in circular microchannels under an imposed constant wall heat flux. The researchers analyzed effects of slip, radius of the plug flow region, and viscous heating on heat transfer and entropy generation. They found that increasing the Brinkman number (Br) (i.e. larger viscous dissipation or smaller wall heat flux) and dimensionless radius of the plug flow region (i.e. greater yield stress or lower pressure gradient) both led to increasing entropy generation and decreasing Nusselt numbers. The temperature difference between fluid and wall was decreased as slip was increased for surface heating. For surface cooling, a similar trend might or might not be observed, decided by the range of the slip coefficient as well as the Brinkman number (Br). The influence of increasing dimensionless radius of the plug flow region was to broaden the temperature profiles for both surface heating and cooling, resulting in enlarging the temperature difference between fluid and wall. Nusselt numbers increased with slip (i.e. smaller friction), while for entropy generation, the reverse was true. A decrease in slip or an increase in radius of the plug flow region resulted in enhancing total irreversibility. They simplified the generalized expressions and results to the familiar Newtonian case as dimensionless radius of the plug flow region approached zero. If slip was set to zero, their results coincided with those of the no-slip condition. Table 1 presents a summary of the aforementioned previous studies on entropy generation in microchannels in the absence of nanofluids.
Summary of previous studies on entropy generation in microchannels in the absence of nanofluids.
Entropy generation in microchannels in the presence of nanofluids
Singh et al. 12 provided a theoretical investigation of the entropy generation analysis due to flow and heat transfer in nanofluids. The researchers considered the most common alumina (Al2O3)-water nanofluids as the model fluid. They took three various diameters of tube in their various regimes because entropy is sensitive to diameter. Those were microchannel (0.1 mm), minichannel (1 mm), and conventional channel (10 mm). They used two different models to represent theoretical and experimental values to consider the influence of thermal conductivity and viscosity. They found that the reduced equation with the help of order of magnitude analysis predicted microchannel and conventional channel entropy generation behavior of nanofluids very well. The alumina–water with high viscosity nanofluids were better coolant for use in minichannels and conventional channels with laminar flow and microchannels and minichannel with turbulent flow. They did not advise to use alumina–water nanofluids with high viscosity in microchannels with laminar flow or minichannels and conventional channels with turbulent flow. Also, there was need to develop low-viscosity alumina–water nanofluids for use in microchannel with laminar flow. They observed that flow friction irreversibility was more significant at lower tube diameter, and thermal irreversibility was more significant at higher tube diameter. Finally, there was an optimum diameter at which the entropy generation rate was the minimum for a given nanofluid for both laminar and turbulent flows.
Employing a validated computer simulation model, Li and Kleinstreuer 13 analyzed entropy generation in trapezoidal microchannels for steady laminar flow of pure water and CuO-water nanofluids. Focusing on MCHS applications, the researchers computed the local and volumetric entropy rates caused by frictional and thermal effects for various coolants, inlet temperatures, Reynolds numbers (Re), and channel ARs. They found that there was an optimal Reynolds number range to operate the system due to the characteristics of the two different entropy sources, both related to the inlet Reynolds number. Microchannels with high ARs had a lower suitable operational Reynolds number range. The employment of nanofluids with very low volume fractions of metal nanoparticles could further minimize entropy generation because of their superior thermal properties. For high heat flux conditions such as micro heat sinks, heat transfer–induced entropy generation was dominant for typical microheating systems while frictional entropy generation became more and more important with the increase in fluid inlet velocity or Reynolds number. Finally, they advised to select microchannel geometries based on minimization of total entropy generation, subject to encountered thermal and hydraulic boundary conditions. Employing certain nanofluids as coolants might further benefit the minimization of entropy generation in MCHSs.
Singh et al. 63 carried out an exergy analysis for nanofluids in microchannels. The researchers chose two microchannels with a diameter of 218 and 303 μm, respectively, due to availability of experimental data. The alumina nanoparticles with particle diameter of 45 nm average size were dispersed in deionized (DI) water. They controlled the stability of these nanofluids by their pH. They chose three concentrations of ϕ = 0.25%, 0.5%, and 1% to observe the volume fraction influence. For the entropy generation analysis, they used Bejan equations for internal flow. They used the order of magnitude method to simplify the equations. Initially, they carried out the analysis with the standard correlations (Dittus–Boelter and Blasius equations) for tube flow for laminar region. Frictional and thermal entropy generation rate ratios were found comparable and neither of them could be neglected. The entropy generation rate ratio was above unity for 218-μm channel, while it was below unity for 303-μm channel. The prediction of entropy generation rate ratio was higher for experimental correlations compared to that of theoretical correlations. The entropy generation number was higher for 303-μm channel and more prone to change with concentration. The thermal part of entropy generation was the major part of total entropy generation for 303-μm channel. Also, the absolute entropy generation was higher for 303-μm channel compared to 218-μm channel.
Focusing on MCHS applications, Li et al. 64 discussed the thermal performance of pure fluid flow (water and EG) as well as various nanofluids (i.e. Al2O3-water and ZnO-EG) with various volume fractions. The researchers illustrated the local and volumetric entropy rates caused by frictional and thermal influences for various coolants, geometries, and operational parameters. The Feng–Kleinstreuer (F-K) thermal conductivity model 65 that consisted of a base fluid static part, kbf, after Maxwell 66 and a new “micro-mixing” part, kmm, that is, knf = kbf + kmm, was adopted in the thermal performance study of nanofluid flow in microchannels. Also, they analyzed and compared two effective nanofluid viscosity models in their study. In addition, they evaluated the friction factor, pressure gradient, pumping power, local heat transfer coefficient, thermal resistance, and entropy generation for various nanofluids. Their experimentally validated computational study provided new physical insight and criteria for design applications toward effective micro-system cooling.
Mah et al. 14 reported an analytical study on the viscous dissipation effect on entropy generation in laminar fully developed forced convection of water–alumina nanofluid in circular microchannels. In the first law analysis, the researchers obtained closed-form solutions of the temperature distributions in the radial direction for the models with and without viscous dissipation term in the energy equation. Their results showed that the heat transfer coefficient decreased with nanoparticle volume fraction largely in the laminar regime of nanofluid flow in microchannel when they took into account the viscous dissipation effect. In the second law analysis, they compared the two models by analyzing their relative deviations in entropy generation for various Reynolds number (Re) and nanoparticle volume fraction (ϕ). When they took into account the viscous dissipation effect, they found that the temperature distribution was prominently affected, and consequently, the entropy generation ascribable to the heat transfer irreversibility was significantly increased. The increase in entropy generation induced by the increase in nanoparticle volume fraction (ϕ) was attributed to the increase in both the thermal conductivity and viscosity of nanofluid that caused augmentation in the heat transfer and fluid friction irreversibilities, respectively. By incorporating the viscous dissipation effect, both thermal performance and exergetic effectiveness for forced convection of nanofluid in microchannels dwindled with nanoparticle volume fraction, contrary to the widespread conjecture that nanofluids had advantage over pure fluid associated with higher overall effectiveness from the aspects of first law and second law of thermodynamics.
Shalchi-Tabrizi and Seyf 15 investigated numerically the effect of using Al2O3-water nanofluids with various volume fractions and particle diameters on generated entropy, hydrodynamic performance, and heat transfer characteristics of a tangential microchannel heat sink (TMHS). Figure 4 shows the schematic diagram of the problem. The cold nanofluid enters the TMHS, and after cooling, the system exits it. The researchers found that considerable heat transfer enhancement was possible when using Al2O3-water nanofluids as coolant, and clearly, the enhancement improved with increasing particle concentration and decreasing particle size. However, using nanofluid had also induced drastic effects on the pumping power that increased with particles’ volume fraction and Reynolds number. The particle size (dp) and Reynolds number (Re) effects on thermal and frictional entropy generation rates for a volume fraction (ϕ) of 4% are shown in Figure 5. It can be seen that the contribution of viscous influences to entropy generation is negligible when compared to the heat transfer contribution. Therefore, overall changes in total entropy generation are greatly determined by thermal influences. It is found that with a decrease in nanoparticle size from dp = 47 to 29 nm, the entropy generation is reduced, while the entropy generation decreases with the increase in Reynolds number. Finally, they found that generated total entropy decreased with increasing volume fraction and Reynolds number and decreasing particle size.
Schematic diagram of the tangential micro heat sink (TMHS) assembly and coordinate system considered by Shalchi-Tabrizi and Seyf.
Thermal and frictional entropy generation rates as a function of Reynolds number (Re) for various particle sizes (dp) and with a volume fraction (ϕ) of 4%.
Sohel et al. 67 discussed analytically various kinds of entropy generations in the circular-shaped microchannel and minichannel using various kinds of nanoparticles and base fluids. The researchers used copper (Cu) and alumina (Al2O3) as the nanoparticles and H2O and EG as the base fluids. They varied the volume fractions of the nanoparticles (ϕ) from 2% to 6%. They analyzed the irreversibility or entropy generation analysis as the function of entropy generation ratio, thermal entropy generation rate, and fluid friction entropy generation rate for these kinds of nanofluids in turbulent flow condition using available correlations. Cu-H2O nanofluid showed the highest decreasing entropy generation rate ratio (36%) compared to these nanofluids flow through the microchannel at ϕ = 6%. The higher thermal conductivity of H2O caused to generate much lower thermal entropy generation rate compared to the EG base fluid. The fluid friction entropy generation rate decreased fruitfully with the increase in volume fraction of the nanoparticles. Cu-H2O and Cu-EG nanofluids gave the maximum decreasing rates of the fluid friction entropy generation rate as 38% and 35%, respectively, at ϕ = 6%. Al2O3-H2O and Al2O3-EG nanofluids showed the maximum decreasing rates as 18% and 16%, respectively, at ϕ = 6%. By comparing microchannel and minichannel heat sinks, they found that smaller diameter showed less entropy generation in case of all nanofluids.
Ting et al. 68 studied analytically the streamwise conduction effect on the entropy generation of low-Péclet-number nanofluid flow in circular MCHSs under exponentially decaying wall heat flux with constant pumping power condition. The researchers developed mathematical models with and without streamwise conduction term in the energy equation and obtained closed-form solutions. Due to the dramatic increase in the effective thermal conductivity of nanofluids, the streamwise conduction influence was justified to be more significant in the nanofluids compared to their base fluids. The streamwise conduction significance, which was prevalent in low-Péclet-number flow regime, was greatly amplified when the volume fraction of nanoparticle was increased. The thermal characteristics of nanofluids were prominently influenced due to the streamwise conduction presence that consequently altered the characteristics of entropy generation in the second law analysis. They observed significant deviations when comparing the cases with and without streamwise conduction. They identified minimal entropy generation in certain range of the low-Péclet-number flow regime, providing an ideal operating condition from the second law aspect for nanofluid flow in microchannels. Also, they investigated the exergetic effectiveness based on the variations on the volume fraction of nanoparticle suspension and the microchannel AR.
Hajialigol et al. 69 studied numerically mixed convection and second law of thermodynamics analysis in a three-dimensional microchannel filled with a nanofluid under a magnetic field. The researchers investigated the temperature fields, variation of horizontal velocity, thermal resistance, pressure drop, Hartmann number, and Reynolds number. Also, they surveyed heat, frictional entropy generation, and magnetic entropy generation in various volume fractions. Analyzing the results of numerical simulations indicated that with increasing Hartmann number (Ha), maximum horizontal velocity along the center line and the inlet and outlet thermal resistances decreased in the microchannel. On the other hand, by enhancing the strength of the imposing magnetic field, heat entropy generation mitigated, while frictional and magnetic ones increased, which was very small compared to heat entropy generation. The ratio of average Nusselt number (Nuavg) to pressure drop was greater than 10. Therefore, the thermal gain of this microchannel fairly dominated the loss of pressure reduction.
Based on the first law and second law of thermodynamics, Ting et al. 70 investigated thermal performance and entropy generation of water–alumina nanofluid flows in porous media embedded in a microchannel under local thermal nonequilibrium condition. The researchers obtained analytical closed-form solutions of two-dimensional temperature distributions for the cases with and without the viscous dissipation term in the energy equation. They derived the thermal nonequilibrium entropy generation function using the differential method. Due to the embedment of the porous medium in the microchannel and the suspension of the nanoparticle in the working fluid, the viscous dissipation influence was magnified significantly, altering thermal characteristics and entropy generation of the system. For the case where the viscous dissipation influence was neglected, total entropy generation and fluid friction irreversibility were overrated, while heat transfer irreversibility was remarkably underestimated. In a low-aspect-ratio microchannel, the suspension of nanoparticles in the fluid decreased the thermodynamic efficiency from the second law point of view. Figure 6 shows the total entropy generation for high-aspect-ratio porous microchannel and the corresponding relative deviations (the value of the total entropy generation for nanofluid divided by the value of the total entropy generation for base fluid (ϕ = 0%)) due to nanoparticle suspensions as a function of Reynolds number. Figure 6 illustrates the existence of the threshold Reynolds number (Reth) due to the balance between the opposite influences of nanoparticle volume fraction (ϕ) on heat transfer irreversibility and fluid friction irreversibility. As the Reynolds number (Re) increased, the contribution of heat transfer irreversibility to total entropy generation decreased, while the contribution of fluid friction irreversibility increased, causing the negative influence of nanoparticle suspension on fluid friction irreversibility to balance out with its positive influence on heat transfer irreversibility at Reth = 100. It was observed that the total entropy generation decreased with Reynolds number (Re) when the Reynolds number (Re) value was extremely low, but the trend was reversed after a short range of Reynolds number (Re), creating a minimum entropy generation point that served as the optimum operating condition of the flow. The optimum Reynolds number (Reopt) associated with minimum entropy generation for nanofluid flow in a porous microchannel was identified to equal 15. With suspension of nanoparticles, the minimum entropy generation point decreased, signifying the advantages of using a nanofluid when the system was operated at optimum condition. From the plot of the corresponding relative deviations (Δ), the existence of a threshold value in Reynolds number at Reth = 100 where the nanoparticle volume fraction (ϕ) influence was marginal was noted. When Re < Reth, suspension of nanoparticles reduced irreversibility in the flow, but the reverse was true when Re > Reth. Therefore, utilization of nanofluids in a high-aspect-ratio microchannel enhanced exergetic effectiveness in low-Reynolds-number flow regime (i.e. when the operating condition of the system was below the Reynolds number threshold value (Reth) = 100). Figure 6 shows that entropy generation in water–alumina nanofluid with ϕ = 4% was decreased with 16% at Re = 5 but increased with 27% at Re = 200. By reducing the nanoparticle size below the threshold value of nanoparticle diameter (dp = 20 nm), entropy generation could be decreased by as much as 73%. The optimum range of porous medium permeability was characterized by image. They observed that effectiveness of the interstitial heat transfer between the solid and fluid phases of the porous medium induced a pronounced influence on the entropy generation, signifying the importance to consider the thermal nonequilibrium condition in the second law performance analysis of porous medium flow.
The total entropy generation for high-aspect-ratio porous microchannel and the corresponding relative deviations due to nanoparticle suspensions as a function of Reynolds number.
Yang et al. 71 presented the numerical simulation of the three-dimensional incompressible steady and laminar fluid flow of a trapezoidal MCHS using nanofluids as a cooling fluid. The researchers discretized Navier–Stokes equations with a conjugate energy equation using the finite volume method. They performed numerical computations for inlet velocity (Win) = 4, 6, and 10 m/s, hydraulic diameter (dh) = 106.66 μm, and heat flux (q) = 200 kW/m2. They demonstrated numerical optimization as a trapezoidal MCHS design that used the combination of a full factorial design and the genetic algorithm method. Three optimal design variables represented the ratio of upper width and lower width of the microchannel (1.2 ≤ α ≤ 3.6), the ratio of the height of the microchannel to the difference between the upper and lower width of the microchannel (0.5 ≤ β ≤ 1.866), and the volume fraction (0% ≤ ϕ ≤ 4%). The dimensionless entropy generation rate of a trapezoidal MCHS was minimized for fixed heat flux and inlet velocity. Their numerical results for the system dimensionless entropy generation rate showed that the system dimensionless friction entropy generation rate increased with Reynolds number (Re). On the contrary, the system dimensionless thermal entropy generation rate decreased with Reynolds number (Re). They showed that the two-phase model (the mixture model) gave higher enhancement than the single-phase model assuming a steadily developing laminar flow. Table 2 presents a summary of the aforementioned previous studies on entropy generation in microchannels in the presence of nanofluids.
Summary of previous studies on entropy generation in microchannels in the presence of nanofluids.
DI: deionized.
Recommendations for future studies
This study will address new points, which will be expected to be the research focus in the coming years. Studying the entropy generation in microchannels can be done using the following:
1. Studying the entropy generation in microchannels of other cross section such as elliptic microchannels. To the best of the author’s knowledge, the study of entropy generation in microchannels of elliptic cross section is not yet tackled in the literature. Only recently, a new interest has been devoted to the elliptical cross section, produced by mechanical fabrication in metallic microchannels for practical applications in MEMS.
Also, the study of the entropy generation in microchannels can be done with the use of different triangular cross sections such as equilateral triangular and right isosceles triangular and “channels of arbitrary shape.” For example, “open channels” of many modern compact and ultra-compact heat exchangers can be properly optimized on the basis of EGM.
2. Studying the entropy generation in microchannels with the use of new working fluids such as nanorefrigerants. Nanorefrigerant is a type of nanofluid where a refrigerant is used as the base fluid. 72 Recently, Dalkiliç and colleagues73,74 presented a review article on nanorefrigerants. To the best of the author’s knowledge, the study of entropy generation in microchannels using nanorefrigerants as the working fluids is not yet tackled in the literature.
Also, the study of the entropy generation in microchannels can be done with the use of magnetic nanofluids (MNF or ferrofluids) that consist of colloidal mixtures of superparamagnetic nanoparticles suspended in a nonmagnetic carrier fluid. MNF constitute a special class of nanofluids, which exhibit both magnetic and fluid properties. The interests in the use of MNF as a heat transfer medium stem from a possibility of controlling its flow and heat transfer process using an external magnetic field. 75
3. For the case of using nanofluids as the working fluids, thermodynamic optimum of microchannels based on entropy generation analysis can be done using different models of thermal conductivity (k) to calculate the Nusselt number (Nu) and hence the entropy generation due to heat transfer. Also, it can be done using various models of dynamic viscosity (μ) to calculate the Reynolds number (Re) and hence the entropy generation due to fluid flow. This is because combining of different models of thermal conductivity (k) with different models of dynamic viscosity (μ) will lead to different results.
4. For the case of using nanofluids as the working fluids, thermodynamic optimum of microchannels based on entropy generation analysis can be done using different kinds of “nanoparticles” as well as different kinds of “base fluid” that are not used before in the literature can be employed (Ag, Au, CuO, diamond, SiO2, TiO2, zirconia (ZrO2), etc.). For example, Yarmand et al. 76 used zirconia (ZrO2) as nanoparticles in macroscale to study entropy generation during turbulent flow of zirconia–water and other nanofluids in a square cross section tube of 10-mm hydraulic diameter with a constant heat flux (q) of 50 kW/m2.
Summary and conclusion
This article provides a comprehensive, up-to-date review in a chronological order on the research progress made on EGM (thermodynamic optimization or finite time thermodynamics). EGM is the method which combines simple models using the most basic concepts of heat transfer, fluid mechanics, and thermodynamics. 1 These simple models are used in the real (irreversible) devices and processes optimization, subject to finite size and finite time constraints. This article is related to using EGM method in microchannels. Examples are drawn from different kinds of applications such as MCHX and MCHS. Finally, some suggestions for future work are presented. The aim of this article is to motivate the researchers to pay more attention to the entropy generation analysis of heat and fluid flow in microchannels to improve the system performance.
Footnotes
Appendix 1
Acknowledgements
The author thanks Monica Toma, Editorial Office, Advances in Mechanical Engineering, Hindawi Publishing Corporation, for inviting him to submit an invited contribution to Advances in Mechanical Engineering with waiving the article processing charges of US$2000 for this invited contribution.
Academic Editor: Anand Thite
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.