**2.1 Fin shape**

Heat exchanger fins are often used in heat exchange devices to increase the heat transfer rate between the heat-exchange surface and the surrounding fluid. Extended surfaces (fins) enhance heat transfer rate by increasing surface area and by inducing turbulent mixing of flow. They can be found in many engineering applications such as the cooling of turbine blades in gas turbine engines, the cooling of electronic components, and different other heat exchange devices used in aerospace, aircraft, chemical processing plants, …, etc. There are different kinds of heat exchanger fins, ranging from relatively simple shapes, like rectangular, cylindrical, annular, tapered or pin fins, to a combination of various geometries. These fins may protrude from either a cylindrical or rectangular base.

parameters as they relate to not only thermal performance but also viscous effects.

Numerous analysis tools are available for determining the thermal performance of heat sinks given a well defined set of design conditions. Convective optimizations are available, such as those presented in Kraus and Bar-Cohen (1995), however, these models assumes a prescribed heat transfer coefficient over the length of the fins which is constant, while in most heat sink applications, hydrodynamic and thermal entrance effects introduce a variable heat transfer coefficient, at least over a portion of the heat sink. The assumption of a constant value of heat transfer coefficient can no longer be prescribed, since the value will depend upon fin spacing and length in the direction of flow. Optimization routines that lead to changes in fin spacing, fin height or fin length also result in changes in the mean heat transfer coefficient and head loss in such a way that iterative procedures are required. While in some instances parametric studies can be undertaken to obtain a relationship between thermal performance and design parameters, a comprehensive design tool should also take into consideration the effect of viscous dissipation and its relationship on thermal performance. The entropy generation associated with heat transfer and frictional effects serve as a direct measure of lost potential for work or in the case of a heat sinks and other finned systems. A modeling approach that establishes a relationship between entropy

Thermodynamic Optimization 33

minimization of the entropy generation associated with heat transfer and fluid friction. All relevant design parameters for plate fin heat sinks, including geometric parameters, heat dissipation, material properties and flow conditions could be simultaneously optimized to characterize a heat sink that minimized entropy generation and in turn results in a minimum operating temperature. The researchers modified Eq. (43) to account for the overall sink resistance rather than the resistance of a single fin using a simple control

<sup>2</sup> . . sin

*gen*

*S*

2

Using Eq. (45), along with the appropriate expressions for the fin resistance, convective heat transfer coefficient, and frictional/drag losses, a model for the entropy generation rate was

Also, they integrated a novel approach for incorporating forced convection through the specification of a fan curve into the optimization procedure, providing a link between optimized design parameters and the system operating point. They presented examples that demonstrated the robust nature of the model for conditions typically found in electronic applications. It was not unusual for a designer to be given an overall maximum heat sink volume. The examples presented in Culham and Muzychka (2001) were assumed to be constrained by a overall maximum volume of 50 mm × 50 mm × 25 mm. In addition, it was assumed that a total heat dissipation of 30 W was uniformly applied over the base plate of the heat sink that had a uniform thickness of 2 mm. Other constraints that were fixed were the thermal conductivity of the heat sink at *k* = 200 W/m.K and the ambient temperature of

Culham and Muzychka (2001) presented several cases that demonstrated the method of entropy generation minimization for sizing plate fin heat sinks. Their examples included single and multi-parameter optimizations. Their results demonstrated the influence of introducing progressively more unconstrained variables into the optimization procedure. The system of non-linear equations for several cases could be solved using numerical procedures like Newton-Raphson solution, contained within many commercially available algebraic software tools. Given the geometric constraints and a uniform heat load to the base plate of the heat sink of 30 W, an optimum number of fins, *N*, was to be determined when *Vf* = 2 m/s, *t* = 1 mm, and *H* = 25 mm. As shown in Table 1, the estimation of the appropriate number of fins was N 29. It was easily seen that decreasing the number of fins led to an increase in the thermal resistance of the heat sink which in turn led to an increase in the temperature excess and a resultant increase in the entropy generation rate. Increasing the number of fins beyond the optimized value would lead to a decrease in the heat sink resistance and temperature excess, but the increase in the head loss associated with fluid

While the optimization procedure estimated the optimum number of fins to be 28.57 the relatively wide range of near minimum entropy generation rate between 20 < *N* < 35, provided designers with a range of options when specifying the appropriate number of fins. In subsequent applications of the optimization method, additional design variables were

*Q R F V*

*T T*

*d f k*

(45)

volume analysis as follows:

developed for an array of parallel plates.

the surrounding air medium at *To* = 25 ◦C or 298 K.

drag would result in an increase in the entropy generation rate.

generation and a fin design parameters, can be used in such a manner that all relevant design conditions combine to produce the best possible thermal sink for the given constraints.

Poulikakos and Bejan (1982) established a theoretical framework for the minimization of entropy generation in forced convection for the design of extended surfaces by the use of the first and second laws of thermodynamics. First, the researchers derived the entropy generation rate formula for a general fin. The entropy generation rate for extended surfaces in external flow with conductive resistance was defined by the following relationship:

$$\dot{S}\_{\text{gen}} = \frac{\dot{\underline{Q}} \, \theta\_b}{T\_{\text{op}}^2} + \frac{F\_d V\_f}{T\_{\text{op}}} \tag{43}$$

The temperature excess of the fin or heat sink (*<sup>b</sup>*) might be related to the overall system thermal resistance:

$$
\partial\_b = \dot{\bar{Q}} \, R\_{fin} \tag{44}
$$

Based on this general result, they developed analytical methods and graphic results for selecting the optimum dimensions of pin fins, rectangular plate fins, plate fins with trapezoidal cross section, and triangular plate fins with rectangular cross section.

Lee and Lin (1995) examined the performance and the entropy generation rate of a fractallike fin under crossflow. This fin type was defined as a fin with subfins repeatedly extending in a fixed way.

Khan et al. (2006) examined the role of cross-sectional shape on entropy generation for several widely used fin cross-sections. The cross-sections examined were circular, elliptical, square, and rectangular. The researchers obtained a general dimensionless expression for the entropy generation rate by considering a control volume around the pin fin including base plate and applying the conservations equations for mass and energy with the entropy balance. They developed the formulation for the dimensionless entropy generation rate in terms of dimensionless variables, including the aspect ratio, Reynolds number, Nusselt number, and the drag coefficient. They examined selected fin geometries for the heat transfer, fluid friction, and the minimum entropy generation rate corresponding to various parameters including axis ratio, aspect ratio, and Reynolds number. Their results clearly indicated that the preferred fin profile was very dependent on these parameters. As the fin became more slender two effects contribute to the reduction in entropy generation number, namely increased surface area that reduced the temperature excess, and a reduction in profile drag which in turn reduced the viscous losses.

#### **2.2 Plate fin arrays**

It is well known that in plate fin type heat exchangers the backmixing and other deviations from plug flow contribute significantly to the inefficiency of the heat exchanger that is important to heat exchangers working in the cryogenic regime.

Culham and Muzychka (2001) presented a procedure that allowed the simultaneous optimization of heat sink design parameters for electronic applications based on a

generation and a fin design parameters, can be used in such a manner that all relevant design conditions combine to produce the best possible thermal sink for the given

Poulikakos and Bejan (1982) established a theoretical framework for the minimization of entropy generation in forced convection for the design of extended surfaces by the use of the first and second laws of thermodynamics. First, the researchers derived the entropy generation rate formula for a general fin. The entropy generation rate for extended surfaces in external flow with conductive resistance was defined by the following relationship:

> 

*T <sup>Q</sup> <sup>S</sup> <sup>b</sup> fd gen* <sup>2</sup>

.

Based on this general result, they developed analytical methods and graphic results for selecting the optimum dimensions of pin fins, rectangular plate fins, plate fins with

Lee and Lin (1995) examined the performance and the entropy generation rate of a fractallike fin under crossflow. This fin type was defined as a fin with subfins repeatedly

Khan et al. (2006) examined the role of cross-sectional shape on entropy generation for several widely used fin cross-sections. The cross-sections examined were circular, elliptical, square, and rectangular. The researchers obtained a general dimensionless expression for the entropy generation rate by considering a control volume around the pin fin including base plate and applying the conservations equations for mass and energy with the entropy balance. They developed the formulation for the dimensionless entropy generation rate in terms of dimensionless variables, including the aspect ratio, Reynolds number, Nusselt number, and the drag coefficient. They examined selected fin geometries for the heat transfer, fluid friction, and the minimum entropy generation rate corresponding to various parameters including axis ratio, aspect ratio, and Reynolds number. Their results clearly indicated that the preferred fin profile was very dependent on these parameters. As the fin became more slender two effects contribute to the reduction in entropy generation number, namely increased surface area that reduced the temperature excess, and a reduction in

It is well known that in plate fin type heat exchangers the backmixing and other deviations from plug flow contribute significantly to the inefficiency of the heat exchanger that is

Culham and Muzychka (2001) presented a procedure that allowed the simultaneous optimization of heat sink design parameters for electronic applications based on a

. .

trapezoidal cross section, and triangular plate fins with rectangular cross section.

The temperature excess of the fin or heat sink (

profile drag which in turn reduced the viscous losses.

important to heat exchangers working in the cryogenic regime.

*T VF*

(43)

*<sup>b</sup>*) might be related to the overall system

*<sup>b</sup> Q Rfin* (44)

constraints.

thermal resistance:

extending in a fixed way.

**2.2 Plate fin arrays** 

minimization of the entropy generation associated with heat transfer and fluid friction. All relevant design parameters for plate fin heat sinks, including geometric parameters, heat dissipation, material properties and flow conditions could be simultaneously optimized to characterize a heat sink that minimized entropy generation and in turn results in a minimum operating temperature. The researchers modified Eq. (43) to account for the overall sink resistance rather than the resistance of a single fin using a simple control volume analysis as follows:

$$\dot{S}\_{gen} = \frac{\dot{Q}^2}{T\_{\omega}^2} R\_{\sin k} + \frac{F\_d V\_f}{T\_{\omega}} \tag{45}$$

Using Eq. (45), along with the appropriate expressions for the fin resistance, convective heat transfer coefficient, and frictional/drag losses, a model for the entropy generation rate was developed for an array of parallel plates.

Also, they integrated a novel approach for incorporating forced convection through the specification of a fan curve into the optimization procedure, providing a link between optimized design parameters and the system operating point. They presented examples that demonstrated the robust nature of the model for conditions typically found in electronic applications. It was not unusual for a designer to be given an overall maximum heat sink volume. The examples presented in Culham and Muzychka (2001) were assumed to be constrained by a overall maximum volume of 50 mm × 50 mm × 25 mm. In addition, it was assumed that a total heat dissipation of 30 W was uniformly applied over the base plate of the heat sink that had a uniform thickness of 2 mm. Other constraints that were fixed were the thermal conductivity of the heat sink at *k* = 200 W/m.K and the ambient temperature of the surrounding air medium at *To* = 25 ◦C or 298 K.

Culham and Muzychka (2001) presented several cases that demonstrated the method of entropy generation minimization for sizing plate fin heat sinks. Their examples included single and multi-parameter optimizations. Their results demonstrated the influence of introducing progressively more unconstrained variables into the optimization procedure. The system of non-linear equations for several cases could be solved using numerical procedures like Newton-Raphson solution, contained within many commercially available algebraic software tools. Given the geometric constraints and a uniform heat load to the base plate of the heat sink of 30 W, an optimum number of fins, *N*, was to be determined when *Vf* = 2 m/s, *t* = 1 mm, and *H* = 25 mm. As shown in Table 1, the estimation of the appropriate number of fins was N 29. It was easily seen that decreasing the number of fins led to an increase in the thermal resistance of the heat sink which in turn led to an increase in the temperature excess and a resultant increase in the entropy generation rate. Increasing the number of fins beyond the optimized value would lead to a decrease in the heat sink resistance and temperature excess, but the increase in the head loss associated with fluid drag would result in an increase in the entropy generation rate.

While the optimization procedure estimated the optimum number of fins to be 28.57 the relatively wide range of near minimum entropy generation rate between 20 < *N* < 35, provided designers with a range of options when specifying the appropriate number of fins. In subsequent applications of the optimization method, additional design variables were

Thermodynamic Optimization 35

with constrained variable optimization. The researchers adapted the method to include a thermal spreading resistance in the overall thermal circuit. Their method characterized the contribution to entropy production of all relevant thermal resistances in the path between source and sink as well as the contribution to viscous dissipation associated with fluid flow at the boundaries of the heat sink. The minimization procedure provided a fast, convenient method for establishing the "best case" design characteristics of plate fin heat sinks given a set of prescribed boundary conditions. They showed that heat sinks made of composite materials containing nonmetallic constituents, with a thermal conductivity as much as an order of magnitude less that typical metallic heat sinks, could provide an effective alternative where performance, cost, and manufacturability were of importance. Also, they showed that the spreading resistance encountered when heat flows from a heat source to the base plate of a heat sink, while significant, could be compensated for by making appropriate

Iyengar and Bar-Cohen (2003) presented a coefficient of performance (*COPT*) analysis for plate fin heat sinks in forced convection and showed to provide a viable technique for combining least-material optimization with the entropy minimization methodology. The *COPT* metric related the heat sink cooling capability to the invested fan pumping work and the thermodynamic work required to manufacture and assemble the heat sink. The proposed optimization methodology maximized the forced convection cooling that could be achieved by a heat sink occupying a specified volume, with a fixed energy investment and entropy generation rate. Also, their study identified the presence of an optimal resource allocation ratio, providing the most favorable distribution of existing energy resources,

Abbassi (2007) investigated the entropy generation in a uniformly heated microchannel heat sink (*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 dimensionless form. Then, average entropy generation rate was utilized as a criterion for assessing the system performance. At the end, the effect of

*S*), group parameter (*Br/*

) on thermal and total entropy generation was

), thermal

between heat sink manufacturing and operation, over a fixed product life cycle.

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

Khan et al. (2009) employed an entropy generation minimization (*EGM*) procedure to optimize the overall performance of microchannel heat sinks. 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 aspect ratio, fin spacing ratio, heat sink material, Knudsen numbers, and accommodation coefficients on the entropy generation rate in the slip flow region. They

influential parameters like, channel aspect ratio (

conductivity ratio (*C*) and porosity (

optimization of MCHS.

design modifications to the heat sink.

introduced into the procedure to simultaneously consider multiple parameters that led to an optimization of the temperature excess and the head loss of the heat sink.

Additional parameters were left unconstrained, like velocity (*Vf*), fin height (*H*), number of fins (*N*), and fin thickness (*t*). Case (ii) examined the influence of relaxing the constraint on free stream velocity prescribed in Case (i) while all other assumed constraints remained unchanged. As shown in Table 1, the optimized number of fins was determined to be *N* 27 and the approach velocity was estimated to be *Vf* = 2.81 m/s for minimum entropy generation. A decrease in the number of fins and an increase in the free stream velocity led to a heat sink with a lower temperature excess but a higher head loss. Overall, the entropy generation rate for this case was lower than in the previous example. Case (iii) examined a three parameter optimization where the constraint on the fin thickness was removed. The results of the optimization gave *N* 38, *Vf* = 3.28 m/s, and *t* = 0.4 mm as shown in Table 1. Further gains had been made in lowering the heat sink temperature excess and head loss that resulted in yet a further decrease in the entropy generation rate. However, the fin thickness might be too thin for practical manufacturing considerations. Finally, none of the variables of interest would be constrained to predetermined values, thus providing a simultaneous optimization of all design variables, including the free stream velocity (*Vf*), the number of fins (*N*), the fin thickness (*t*), and the fin height (*H*). Their results of the optimization gave *N* 19, *Vf* = 1.21 m/s, *t* = 1.6 mm, and *H* = 122 mm. Once again a more optimal solution had been found. While the approach presented provided an optimized heat sink, the fin height exceeded the maximum allowable height of 25 mm predicated by the board-to-board spacing.

Moreover, it was important to note, that as more variables became unconstrained, the system was progressively seeking a more optimal design. For instance, in cases (ii) and (iii), although the fin count increased, the fin thickness decreased, leading not only to a thermally more efficient design, but also a system that used less material. Finally, one might introduce additional constraints as needed that limited the temperature excess or the mass of the heat sink. Their method outlined was also applicable to fin arrays used in heat exchangers.


Table 1. Optimized Conditions for All Test Cases.

Their model was shown to converge to a unique solution that gave the optimized design conditions for the imposed problem constraints.

The specification and design of heat sinks for electronic applications is not easily accomplished through the use of conventional thermal analysis tools because "optimized" geometric and boundary conditions are not known a priori.

Culham et al. (2007) presented an analytical model for calculating the best possible design parameters for plate fin heat sinks using an entropy generation minimization procedure

introduced into the procedure to simultaneously consider multiple parameters that led to an

Additional parameters were left unconstrained, like velocity (*Vf*), fin height (*H*), number of fins (*N*), and fin thickness (*t*). Case (ii) examined the influence of relaxing the constraint on free stream velocity prescribed in Case (i) while all other assumed constraints remained unchanged. As shown in Table 1, the optimized number of fins was determined to be *N* 27 and the approach velocity was estimated to be *Vf* = 2.81 m/s for minimum entropy generation. A decrease in the number of fins and an increase in the free stream velocity led to a heat sink with a lower temperature excess but a higher head loss. Overall, the entropy generation rate for this case was lower than in the previous example. Case (iii) examined a three parameter optimization where the constraint on the fin thickness was removed. The results of the optimization gave *N* 38, *Vf* = 3.28 m/s, and *t* = 0.4 mm as shown in Table 1. Further gains had been made in lowering the heat sink temperature excess and head loss that resulted in yet a further decrease in the entropy generation rate. However, the fin thickness might be too thin for practical manufacturing considerations. Finally, none of the variables of interest would be constrained to predetermined values, thus providing a simultaneous optimization of all design variables, including the free stream velocity (*Vf*), the number of fins (*N*), the fin thickness (*t*), and the fin height (*H*). Their results of the optimization gave *N* 19, *Vf* = 1.21 m/s, *t* = 1.6 mm, and *H* = 122 mm. Once again a more optimal solution had been found. While the approach presented provided an optimized heat sink, the fin height exceeded the maximum allowable height of 25 mm predicated by the

Moreover, it was important to note, that as more variables became unconstrained, the system was progressively seeking a more optimal design. For instance, in cases (ii) and (iii), although the fin count increased, the fin thickness decreased, leading not only to a thermally more efficient design, but also a system that used less material. Finally, one might introduce additional constraints as needed that limited the temperature excess or the mass of the heat sink. Their method outlined was also applicable to fin arrays used in heat exchangers.

(i) 28.57 11.51 5.62 2.0 1.0 25 0.00435 (ii) 26.77 9.49 7.02 2.81 1.0 25 0.00402 (iii) 38.14 8.66 5.78 3.28 0.4 25 0.00370 (iv) 19.07 7.20 1.90 1.21 1.6 122 0.00290

Their model was shown to converge to a unique solution that gave the optimized design

The specification and design of heat sinks for electronic applications is not easily accomplished through the use of conventional thermal analysis tools because "optimized"

Culham et al. (2007) presented an analytical model for calculating the best possible design parameters for plate fin heat sinks using an entropy generation minimization procedure

(mmH2O) *Vf* (m/s) *t* (mm) *H* (mm) *<sup>S</sup> gen*

.

(W/◦C)

*P*

optimization of the temperature excess and the head loss of the heat sink.

board-to-board spacing.

Case *N* 

*b* (◦C)

Table 1. Optimized Conditions for All Test Cases.

conditions for the imposed problem constraints.

geometric and boundary conditions are not known a priori.

with constrained variable optimization. The researchers adapted the method to include a thermal spreading resistance in the overall thermal circuit. Their method characterized the contribution to entropy production of all relevant thermal resistances in the path between source and sink as well as the contribution to viscous dissipation associated with fluid flow at the boundaries of the heat sink. The minimization procedure provided a fast, convenient method for establishing the "best case" design characteristics of plate fin heat sinks given a set of prescribed boundary conditions. They showed that heat sinks made of composite materials containing nonmetallic constituents, with a thermal conductivity as much as an order of magnitude less that typical metallic heat sinks, could provide an effective alternative where performance, cost, and manufacturability were of importance. Also, they showed that the spreading resistance encountered when heat flows from a heat source to the base plate of a heat sink, while significant, could be compensated for by making appropriate design modifications to the heat sink.

Iyengar and Bar-Cohen (2003) presented a coefficient of performance (*COPT*) analysis for plate fin heat sinks in forced convection and showed to provide a viable technique for combining least-material optimization with the entropy minimization methodology. The *COPT* metric related the heat sink cooling capability to the invested fan pumping work and the thermodynamic work required to manufacture and assemble the heat sink. The proposed optimization methodology maximized the forced convection cooling that could be achieved by a heat sink occupying a specified volume, with a fixed energy investment and entropy generation rate. Also, their study identified the presence of an optimal resource allocation ratio, providing the most favorable distribution of existing energy resources, between heat sink manufacturing and operation, over a fixed product life cycle.

Abbassi (2007) investigated the entropy generation in a uniformly heated microchannel heat sink (*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 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 like, channel aspect ratio (*S*), group parameter (*Br/*), thermal conductivity ratio (*C*) and porosity () 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.

Khan et al. (2009) employed an entropy generation minimization (*EGM*) procedure to optimize the overall performance of microchannel heat sinks. 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 aspect ratio, fin spacing ratio, heat sink material, Knudsen numbers, and accommodation coefficients on the entropy generation rate in the slip flow region. They

Thermodynamic Optimization 37

exchanger. They adopted the classical correlations of the heat transfer and the flow friction to avoid solving the differential equations. As a result, the computation burden of *DPM* became significantly less than that of the Computational Fluid Dynamics method. They performed the optimal design of a *PFHE* based on the *DPM* with the entropy generation minimization taken into consideration. They employed the genetic algorithm to conduct the optimization due to its robustness in dealing with complicated problems. The fin type and fin geometry were selected optimally from a customized fin database. The *PFHE* included in an environmental control system was designed by using the proposed approach in their study. Finally, They evaluated the cooling performance of the optimal *PFHE* under both dry

Galvis and Culham (2010) used the entropy generation minimization (*EGM*) method to find the optimum channel dimensions in micro heat exchangers with a uniform heat flux. With this approach, pressure drop and heat transfer in the micro channels were considered simultaneously during the optimization analysis. The researchers developed a computational model to find the optimum channel depth knowing other channel geometry dimensions and coolant inlet properties. Their assumptions were laminar and both hydrodynamically and thermally fully developed flow, and incompressible. However, they introduced the Hagenbach factor (*K*) to take into account the developing length effect in the friction losses. The Hagenbach factor (*K*) for rectangular channels obtained by Steinke and

23 4 5 K 0.6796 1.2197 3.3089 -9.5921 8.9089 2.9959 *<sup>S</sup> SS S S*

The micro channels were assumed to have an isothermal or isoflux boundary condition, non-slip flow, and fluid properties had dependency on temperature accordingly. For these particular case studies, the pressure drop and heat transfer coefficient for the isothermal boundary condition is lower than the isoflux case. As the channel size decreased, they found higher heat transfer coefficient and pressure drop. The optimum channel geometry that

Rao and Patel (2010) discussed the use of particle swarm optimization (*PSO*) algorithm for thermodynamic optimization of a cross flow plate-fin heat exchanger. The researchers considered minimization of total number of entropy generation units for specific heat duty requirement under given space restrictions, minimization of total volume, and minimization of total annual cost as objective functions and treated individually. Based on the applications, they considered heat exchanger length, fin frequency, numbers of fin layers, lance length of fin, fin height and fin thickness or various flow length of the heat exchanger for optimization. They included heat duty requirement constraint in the procedure. Also, they presented two application examples to demonstrate the effectiveness and accuracy of the proposed algorithm. They validated the results of optimization using *PSO* by comparing with those obtained by using genetic algorithm (*GA*). In addition, they carried out parametric analysis to demonstrate the influence of heat exchanger dimensions on the optimum solution. Moreover, they presented the influence of variation of *PSO* parameters

Ahmadi et al. (2011) conducted a thermal modeling for optimal design of compact heat

exchangers to minimize cost and entropy generation. The researchers applied an

 

 (46)

> - *NTU*

on convergence and optimum value of the objective.

minimized the entropy generation rate tended to be a deep, narrow channel.

and wet conditions.

Kandlikar (2006) as follows:

used analytical/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 microchannel heat sinks.

The thermal design of plate fin heat sinks can benefit from optimization procedures where all design variables are simultaneously prescribed, ensuring the best thermodynamic and air flow characteristic possible. While a cursory review of the thermal network established between heat sources and sinks in typical plate fin heat sinks would indicate that the film resistance at the fluid-solid boundary dominates, it is shown that the effects of other resistance elements, such as the spreading resistance and the material resistance, although of lesser magnitude, play an important role in the optimization and selection of heat sink design conditions.

Zhou et al. (2009) proposed the multi-parameter constrained optimization procedure integrating the design of experiments (*DOE*), response surface models (*RSM*), genetic algorithm (*GA*), mixed integer optimization (*MOST*), and computational fluid dynamics (*CFD*) to design the plate finned heat sinks by minimizing their rates of entropy generation. The results of three cases demonstrated that the combination optimization algorithm was feasible. In these cases, the overall rate of entropy generation decreased as the result of introducing the additional constrained variables into the optimization procedure. As a result, the general thermal and fluid performance of the heat sink was dramatically improved.

Based on the results derived by the optimization, the researchers investigated the overall thermal and fluid performance of the plate finned heat sinks with both side and top bypass flow. Also, they established two correlations describing Nusselt number and friction factor, as the functions of geometrical and operational parameters, by means of the multivariate non-linear regression analysis. They deduced the specific expressions to compute the thermal resistance and the rate of entropy generation.

Ganzarollia, and Altemania (2010) performed the thermal design of a counterflow heat exchanger using air as the working fluid with two distinct goals: minimum inlet temperature difference and minimum number of entropy generation units. The researchers constituted the heat exchanger by a double-finned conductive plate closed by adiabatic walls at the fin tips on both sides. The cold and hot air flows were considered in the turbulent regime, driven by a constant pressure head. The thermal load was constant, and an optimization was performed in order to obtain the optimum fin spacing and thickness, according to the two design criteria. They employed a computer program to evaluate the optimum conditions based on correlations from the literature. They compared the results obtained from both design criteria to each other. They performed a scale analysis considering the first design goal and compared the corresponding dimensionless parameters with the results from the correlations.

Zhang et al. (2010) developed a general three-dimensional distributed parameter model (*DPM*) for designing the plate-fin heat exchanger (*PFHE*). The proposed model that allowed for the varying local fluid thermophysical properties inside the flow path could be applied for both dry and wet working conditions by using the uniform enthalpy equations. The researchers generated the grids in the *DPM* to match closely the flow passage of the heat

used analytical/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

The thermal design of plate fin heat sinks can benefit from optimization procedures where all design variables are simultaneously prescribed, ensuring the best thermodynamic and air flow characteristic possible. While a cursory review of the thermal network established between heat sources and sinks in typical plate fin heat sinks would indicate that the film resistance at the fluid-solid boundary dominates, it is shown that the effects of other resistance elements, such as the spreading resistance and the material resistance, although of lesser magnitude, play an important role in the optimization and selection of heat sink

Zhou et al. (2009) proposed the multi-parameter constrained optimization procedure integrating the design of experiments (*DOE*), response surface models (*RSM*), genetic algorithm (*GA*), mixed integer optimization (*MOST*), and computational fluid dynamics (*CFD*) to design the plate finned heat sinks by minimizing their rates of entropy generation. The results of three cases demonstrated that the combination optimization algorithm was feasible. In these cases, the overall rate of entropy generation decreased as the result of introducing the additional constrained variables into the optimization procedure. As a result, the general thermal and fluid performance of the heat sink was dramatically

Based on the results derived by the optimization, the researchers investigated the overall thermal and fluid performance of the plate finned heat sinks with both side and top bypass flow. Also, they established two correlations describing Nusselt number and friction factor, as the functions of geometrical and operational parameters, by means of the multivariate non-linear regression analysis. They deduced the specific expressions to compute the

Ganzarollia, and Altemania (2010) performed the thermal design of a counterflow heat exchanger using air as the working fluid with two distinct goals: minimum inlet temperature difference and minimum number of entropy generation units. The researchers constituted the heat exchanger by a double-finned conductive plate closed by adiabatic walls at the fin tips on both sides. The cold and hot air flows were considered in the turbulent regime, driven by a constant pressure head. The thermal load was constant, and an optimization was performed in order to obtain the optimum fin spacing and thickness, according to the two design criteria. They employed a computer program to evaluate the optimum conditions based on correlations from the literature. They compared the results obtained from both design criteria to each other. They performed a scale analysis considering the first design goal and compared the corresponding dimensionless

Zhang et al. (2010) developed a general three-dimensional distributed parameter model (*DPM*) for designing the plate-fin heat exchanger (*PFHE*). The proposed model that allowed for the varying local fluid thermophysical properties inside the flow path could be applied for both dry and wet working conditions by using the uniform enthalpy equations. The researchers generated the grids in the *DPM* to match closely the flow passage of the heat

overall performance of microchannel heat sinks.

thermal resistance and the rate of entropy generation.

parameters with the results from the correlations.

design conditions.

improved.

exchanger. They adopted the classical correlations of the heat transfer and the flow friction to avoid solving the differential equations. As a result, the computation burden of *DPM* became significantly less than that of the Computational Fluid Dynamics method. They performed the optimal design of a *PFHE* based on the *DPM* with the entropy generation minimization taken into consideration. They employed the genetic algorithm to conduct the optimization due to its robustness in dealing with complicated problems. The fin type and fin geometry were selected optimally from a customized fin database. The *PFHE* included in an environmental control system was designed by using the proposed approach in their study. Finally, They evaluated the cooling performance of the optimal *PFHE* under both dry and wet conditions.

Galvis and Culham (2010) used the entropy generation minimization (*EGM*) method to find the optimum channel dimensions in micro heat exchangers with a uniform heat flux. With this approach, pressure drop and heat transfer in the micro channels were considered simultaneously during the optimization analysis. The researchers developed a computational model to find the optimum channel depth knowing other channel geometry dimensions and coolant inlet properties. Their assumptions were laminar and both hydrodynamically and thermally fully developed flow, and incompressible. However, they introduced the Hagenbach factor (*K*) to take into account the developing length effect in the friction losses. The Hagenbach factor (*K*) for rectangular channels obtained by Steinke and Kandlikar (2006) as follows:

$$\text{K = } 0.6796 + 1.2197a\_{\text{S}} + 3.3089a\_{\text{s}}^2 \text{-} 9.5921a\_{\text{s}}^3 + 8.9089a\_{\text{s}}^4 + 2.9959a\_{\text{s}}^5 \tag{46}$$

The micro channels were assumed to have an isothermal or isoflux boundary condition, non-slip flow, and fluid properties had dependency on temperature accordingly. For these particular case studies, the pressure drop and heat transfer coefficient for the isothermal boundary condition is lower than the isoflux case. As 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.

Rao and Patel (2010) discussed the use of particle swarm optimization (*PSO*) algorithm for thermodynamic optimization of a cross flow plate-fin heat exchanger. The researchers considered minimization of total number of entropy generation units for specific heat duty requirement under given space restrictions, minimization of total volume, and minimization of total annual cost as objective functions and treated individually. Based on the applications, they considered heat exchanger length, fin frequency, numbers of fin layers, lance length of fin, fin height and fin thickness or various flow length of the heat exchanger for optimization. They included heat duty requirement constraint in the procedure. Also, they presented two application examples to demonstrate the effectiveness and accuracy of the proposed algorithm. They validated the results of optimization using *PSO* by comparing with those obtained by using genetic algorithm (*GA*). In addition, they carried out parametric analysis to demonstrate the influence of heat exchanger dimensions on the optimum solution. Moreover, they presented the influence of variation of *PSO* parameters on convergence and optimum value of the objective.

Ahmadi et al. (2011) conducted a thermal modeling for optimal design of compact heat exchangers to minimize cost and entropy generation. The researchers applied an - *NTU*

Thermodynamic Optimization 39

employed entropy generation and exergy loss to investigate a multiport serpentine slab *MCHX* with ethylene glycol-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 like 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

For heat transfer enhancement, pin fins are widely used as effective elements. For this purpose, extensive work is being carried out to choose and optimize pin fins for different applications. Any optimization procedure would lead to desirable results only if the parallel pressure drop and heat transfer are considered. Pin fin arrays are another popular geometry used in electronics cooling. Pin fins are attractive as a result of their ability to operate easily

First, Lin and Lee (1997) conducted the second law analysis on a pin-fin array under crossflow to evaluate the entropy generation rate. Increasing the crossflow fluid velocity enhancing the heat transfer rate and hence, reducing the heat transfer irreversibility. Nevertheless, owing to the simultaneous increase in drag force exerting on the fin bodies, the hydrodynamic irreversibility increased also. An optimal Reynolds number thereby existed over wide operating conditions. The researchers searched the optimal design/operational conditions on the basis of entropy generation minimization. Also, they

Şara et al. (2001) presented heat transfer and friction characteristics, and the second law analysis of the convective heat transfer through a rectangular channel with square crosssectional pin fins attached over a fiat surface. The researchers used different clearance ratios and interfin distance ratios and determined optimum pin-fin arrays that minimized entropy generation. They found that average Nusselt number based on the projected area decreased with increasing clearance ratio and interfin distance ratio, whereas average Nusselt number based on the total heat transfer area increased with increasing interfin distance ratio and with decreasing clearance ratio. Also, they found that the friction factor to decrease with increasing clearance ratio and interfin distance ratio. They obtained smaller entropy generation numbers at lower Reynolds number, higher clearance ratio, and higher interfin

Khan et al. (2005) applied an entropy generation minimization (EGM) technique as a unique measure to study the thermodynamic losses caused by heat transfer and pressure drop in cylindrical pin-fin heat sinks. The researchers obtained a general expression for the entropy generation rate by considering the whole heat sink as a control volume and applying the conservation equations for mass and energy with the entropy balance. They used analytical/empirical correlations for heat transfer coefficients and friction factors in the optimization model, where the characteristic length was used as the diameter of the pin and reference velocity used in Reynolds number and pressure drop was based on the minimum free area available for the fluid flow. They studied both in-line and staggered arrangements and compared their relative performance on the basis of equal overall volume of heat sinks.

made the comparison of the staggered and the in-line pin-fm alignments.

between the predicted and the measured results.

in multi-directional fluid streams.

**2.3 Pin fins** 

spacing ratio.

method for estimation of the heat exchanger pressure drop, and effectiveness. Fin pitch, fin height, fin offset length, cold stream flow length, no-flow length, and hot stream flow length were considered as six decision variables. They applied fast and elitist nondominated sorting genetic algorithm (i.e., nondominated sorting genetic algorithm II) to minimize the entropy generation units and the total annual cost (sum of initial investment and operating and maintenance costs) simultaneously. The results for Pareto-optimal front clearly revealed the conflict between two objective functions, the number of entropy generation units (*Ns*) and the total annual cost (*Ctotal*). It revealed that any geometrical changes that decreased the number of entropy generation units, led to an increase in the total annual cost and vice versa. Moreover, they derived an equation for the number of entropy generation units versus the total annual cost for the Pareto curve for prediction of the optimal design of the plate fin heat exchanger as follows:

$$\text{C}\_{\text{total}} \left( \text{\\$} \right) = \frac{-2.819 \text{N}\_s^3 - 4.311 \text{N}\_s^2 + 1.728 \text{N}\_s - 0.04891}{\text{N}\_s^2 + 21.84 \text{N}\_s - 1.867} \text{ x} \\ \text{10,000} \quad 0.0939 < \text{N}\_s < 0.13 \text{ (47)}$$

Considering a numerical value for the number of entropy generation units in the range 0.0939 < *N*s < 0.13 provided the minimum total annual cost for that optimal point along with other optimal design parameters. Also, optimization of heat exchangers based on considering exergy destruction revealed that irreversibilities, like pressure drop and high temperature difference between cold and hot streams, played a key issue in exergy destruction. Thus, more efficient heat exchanger led to have a heat exchanger with higher total cost rate. At the end, the sensitivity analysis of change in the optimum number of entropy generation units and the total annual cost with change in the decision variables of the plate fin heat exchanger was also performed, and the results were reported.

Shuja and Zubair (2011) presented a detailed second-law based thermoeconomic optimization for a finned heat sink array. This involved including costs associated with material and irreversible losses due to heat transfer and pressure drop. The researchers optimized the effect of important physical, geometrical and unit cost parameters on the overall finned array for some typical operating conditions that were representative of electronic cooling applications. They presented the cost optimized results in terms of different parameters for a finned system. Furthermore, they explained the methodology of obtaining optimum design parameters for a finned heat sink system that would result in minimum total cost.

Gielen et al. (2011) discussed the use of second law based cost functions in plate fin heat sink design. The researchers proposed and compared a new entropy-based cost function with existing heat sink cost functions. A case study of a plate fin heat sink pointed out that their newly developed cost function offered a heat sink that was more than twice as efficient as a heat sink designed with the traditional thermal resistance minimization objective. The influences of this new heat sink design on data center cooling systems were considered and found to be significantly improving the system efficiency and waste heat recovery.

Al-Obaidi (2011) 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 ethylene glycol-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 like 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.
