**2.3 Pin fins**

(47)

38 Heat Exchangers – Basics Design Applications

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

ss s total <sup>2</sup> <sup>s</sup>


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

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

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

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

found to be significantly improving the system efficiency and waste heat recovery.

plate fin heat exchanger as follows:

minimum total cost.

3 2

s s

N 21.84N - 1.867

the plate fin heat exchanger was also performed, and the results were reported.

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 in multi-directional fluid streams.

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

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 spacing ratio.

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.

Thermodynamic Optimization 41

analysis for three different models of pin-fin heat sinks. The models were different in cross section area. These cross section areas were circle, horizontal ellipse, and vertical ellipse. Reference velocity used in Reynolds number and pressure drop was based on the minimum free area available for the fluid flow. As expected, the pressure drop and entropy generation increased with the rise of frontal velocity. Also, they investigated in-line arrangement of fins for numerical analysis and compared their relative performance. Finally, they compared the performance of these three models from the views point of heat transfer and total entropy generation rate. The elliptic pin fin showed the lowest pressure drops. Whereas, the circular geometry appeared as the best from the view point of the total entropy generation rate for low approach velocities and the elliptical geometry was the next favorable geometry from the view point of total entropy generation rate for higher approach velocities. Eventually, vertical elliptic fins showed the highest pressure drop and entropy generation among these

Su et al. (2011) studied theoretically the entropy generation during heat transfer of a pin fin array in channels with lateral ejection holes for a turbine blade. The researchers established the entropy generation model based on the second-law analysis. They analyzed the distribution of the entropy generation due to heat transfer and fluid friction irreversibility respectively and made a comparison for three typical short pin fin channels. The entropy generation number component due to heat transfer decreased while *Red* increased, while the component due to fluid friction increased with the increase of *Red*. The entropy generation number (*Ns*) reached minimum when the two components met and the corresponding Reynolds number (*Red*) was optimal. The ejection holes affected the energy lost of the working fluid. For the three cases studied in this work, case b with short ejection holes gave the best comprehensive thermal performance with comparison to cases with no and long ejection holes. They suggested that their results would be helpful for the design of the heat

This chapter provides a comprehensive, up-to-date review in a chronological order on the research progress made on entropy generation minimization (thermodynamic optimization, or finite time thermodynamics). *EGM* is the method which combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics (Bejan, 1982a). These simple models are used in the real (irreversible) devices and processes optimization, subject to finite‐size and finite‐time constraints. The current review is related to using *EGM* method in heat exchangers for both internal structure and external structure. Examples are drawn from different kinds of applications: parallel flow, counterflow, crossflow, phasechange heat exchanger optimizations, as well as optimization of internal enhancement. Attention is also gives to micro systems such as microchannel heat exchanger (*MCHX*).

The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants program. Also, we

want to thank the Editor, Prof. Jovan Mitrovic, for inviting us to prepare this chapter.

different geometries.

**3. Conclusion** 

**4. Acknowledgment** 

dissipation of pin fin heat exchangers.

The details of the necessary models for heat transfer and drag might be found in Khan et al. (2005) along with the general control volume analysis. It was shown that all relevant design parameters for pin-fin heat sinks, including geometric parameters, material properties and flow conditions could be simultaneously optimized.

Khan et al. (2008) applied an entropy generation minimization (*EGM*) method to study the thermodynamic losses caused by heat transfer and pressure drop for the fluid in a cylindrical pin-fin heat sink and bypass flow regions. The researchers obtained a general expression for the entropy generation rate by considering control volumes around the heat sink and bypass regions. The conservation equations for mass and energy with the entropy balance were applied in both regions. Inside the heat sink, analytical/empirical correlations were used for heat transfer coefficients and friction factors, where the reference velocity used in the Reynolds number and the pressure drop was based on the minimum free area available for the fluid flow. In bypass regions theoretical models, based on laws of conservation of mass, momentum, and energy, were used to predict flow velocity and pressure drop. They studied both in-line and staggered arrangements and compared their relative performance to the same thermal and hydraulic conditions. Also, they performed a parametric study to show the effects of bypass on the overall performance of heat sinks.

Sahiti et al. (2008) derived experimentally the heat transfer and pressure drop characteristics of a double-pipe pin fin heat exchanger. The researchers used the empirical correlations previously validated with their experimental data to develop a mathematical model for the optimization of the actual heat exchanger. They developed the optimization model on the basis of the entropy generation minimization for various heat exchanger flow lengths and various pin lengths. They derived the conclusions on the basis of the behavior of the entropy generation number (*Ns*) as a function of the Reynolds number (*Re*). They showed that not all definition forms for the entropy generation number led to the right conclusions.

Nwachukwu and Onyegegbu (2009) derived an expression for the optimum pin fin dimension on exergy basis for a high temperature exchanger employing pin fins. Their result was different from that obtained by Poulikakos and Bejan (1982) for a low temperature heat recovery application. In addition, the researchers established a simple relation between the amounts the base temperature of the optimized pin fin was raised for a range of absorptive coating values. Employing this relation, if the absorptivity of the coating, the plate emissivity, the number of protruding fins, and some area and fluid parameters were known, they obtained immediately the corresponding value for the base temperature of the fin. Their analysis showed that the thermal performance of the exchanger improved substantially with a high absorptivity coating hence could be seen as a heat transfer enhancement feature of the exchanger operating with radiation dominance.

Kamali et al. (2010) used numerical analysis to investigate entropy generation for array of pin-fin heat sink. Technique was applied to study the thermodynamic losses caused by heat transfer and pressure drop in 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 and empirical correlations for heat transfer coefficients and friction factors in the numerical modeling. Also, they studied heat transfer and pressure drop effects in entropy generation in control volume over pin-fins. They used numerical

The details of the necessary models for heat transfer and drag might be found in Khan et al. (2005) along with the general control volume analysis. It was shown that all relevant design parameters for pin-fin heat sinks, including geometric parameters, material properties and

Khan et al. (2008) applied an entropy generation minimization (*EGM*) method to study the thermodynamic losses caused by heat transfer and pressure drop for the fluid in a cylindrical pin-fin heat sink and bypass flow regions. The researchers obtained a general expression for the entropy generation rate by considering control volumes around the heat sink and bypass regions. The conservation equations for mass and energy with the entropy balance were applied in both regions. Inside the heat sink, analytical/empirical correlations were used for heat transfer coefficients and friction factors, where the reference velocity used in the Reynolds number and the pressure drop was based on the minimum free area available for the fluid flow. In bypass regions theoretical models, based on laws of conservation of mass, momentum, and energy, were used to predict flow velocity and pressure drop. They studied both in-line and staggered arrangements and compared their relative performance to the same thermal and hydraulic conditions. Also, they performed a parametric study to show the effects of bypass on the overall performance of heat sinks.

Sahiti et al. (2008) derived experimentally the heat transfer and pressure drop characteristics of a double-pipe pin fin heat exchanger. The researchers used the empirical correlations previously validated with their experimental data to develop a mathematical model for the optimization of the actual heat exchanger. They developed the optimization model on the basis of the entropy generation minimization for various heat exchanger flow lengths and various pin lengths. They derived the conclusions on the basis of the behavior of the entropy generation number (*Ns*) as a function of the Reynolds number (*Re*). They showed that not all

Nwachukwu and Onyegegbu (2009) derived an expression for the optimum pin fin dimension on exergy basis for a high temperature exchanger employing pin fins. Their result was different from that obtained by Poulikakos and Bejan (1982) for a low temperature heat recovery application. In addition, the researchers established a simple relation between the amounts the base temperature of the optimized pin fin was raised for a range of absorptive coating values. Employing this relation, if the absorptivity of the coating, the plate emissivity, the number of protruding fins, and some area and fluid parameters were known, they obtained immediately the corresponding value for the base temperature of the fin. Their analysis showed that the thermal performance of the exchanger improved substantially with a high absorptivity coating hence could be seen as a heat

definition forms for the entropy generation number led to the right conclusions.

transfer enhancement feature of the exchanger operating with radiation dominance.

Kamali et al. (2010) used numerical analysis to investigate entropy generation for array of pin-fin heat sink. Technique was applied to study the thermodynamic losses caused by heat transfer and pressure drop in 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 and empirical correlations for heat transfer coefficients and friction factors in the numerical modeling. Also, they studied heat transfer and pressure drop effects in entropy generation in control volume over pin-fins. They used numerical

flow conditions could be simultaneously optimized.

analysis for three different models of pin-fin heat sinks. The models were different in cross section area. These cross section areas were circle, horizontal ellipse, and vertical ellipse. Reference velocity used in Reynolds number and pressure drop was based on the minimum free area available for the fluid flow. As expected, the pressure drop and entropy generation increased with the rise of frontal velocity. Also, they investigated in-line arrangement of fins for numerical analysis and compared their relative performance. Finally, they compared the performance of these three models from the views point of heat transfer and total entropy generation rate. The elliptic pin fin showed the lowest pressure drops. Whereas, the circular geometry appeared as the best from the view point of the total entropy generation rate for low approach velocities and the elliptical geometry was the next favorable geometry from the view point of total entropy generation rate for higher approach velocities. Eventually, vertical elliptic fins showed the highest pressure drop and entropy generation among these different geometries.

Su et al. (2011) studied theoretically the entropy generation during heat transfer of a pin fin array in channels with lateral ejection holes for a turbine blade. The researchers established the entropy generation model based on the second-law analysis. They analyzed the distribution of the entropy generation due to heat transfer and fluid friction irreversibility respectively and made a comparison for three typical short pin fin channels. The entropy generation number component due to heat transfer decreased while *Red* increased, while the component due to fluid friction increased with the increase of *Red*. The entropy generation number (*Ns*) reached minimum when the two components met and the corresponding Reynolds number (*Red*) was optimal. The ejection holes affected the energy lost of the working fluid. For the three cases studied in this work, case b with short ejection holes gave the best comprehensive thermal performance with comparison to cases with no and long ejection holes. They suggested that their results would be helpful for the design of the heat dissipation of pin fin heat exchangers.

### **3. Conclusion**

This chapter provides a comprehensive, up-to-date review in a chronological order on the research progress made on entropy generation minimization (thermodynamic optimization, or finite time thermodynamics). *EGM* is the method which combines into simple models the most basic concepts of heat transfer, fluid mechanics, and thermodynamics (Bejan, 1982a). These simple models are used in the real (irreversible) devices and processes optimization, subject to finite‐size and finite‐time constraints. The current review is related to using *EGM* method in heat exchangers for both internal structure and external structure. Examples are drawn from different kinds of applications: parallel flow, counterflow, crossflow, phasechange heat exchanger optimizations, as well as optimization of internal enhancement. Attention is also gives to micro systems such as microchannel heat exchanger (*MCHX*).

### **4. Acknowledgment**

The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants program. Also, we want to thank the Editor, Prof. Jovan Mitrovic, for inviting us to prepare this chapter.

Thermodynamic Optimization 43

entropy generation rate due to viscous dissipation, W/K

*To* ambient temperature or dead-state temperature, K

*U* overall heat transfer coefficient, W/m2.K

*Ys* heat exchange reversibility norm (HERN)

*cp* , m2/s

*Vf* free stream or approach velocity, m/s

*S\** entropy generation number ratio

*St* Stanton number *T* temperature, K

*t* fin thickness, m

*V* velocity, m/s

difference

 entropy flux *η* efficiency

*Π* Reduced period *θ* blade angle

density, kg/m3

capacity ratios

*0* without augmentation

*h* hydraulic, hot stream

*max* maximum value

*Tr* inlet temperature ratio *T∞* environment temperature, K

*v* specific volume, m3/kg *wC* channel width, m

thermal diffusivity = *k/*

effectiveness, porosity

*ηII* exergy recovery index *Λ* Reduced length

dynamic viscosity, kg/m.s

*<sup>b</sup>* temperature excess of the fin, (*Tb* − *T∞*)

dimensionless temperature difference

entropy generation distribution factor

kinematic viscosity, m2/s

*<sup>S</sup>* channel aspect ratio = *H/wC* 

irreversibility distribution ratio

*S v* .

**Greek** 

**Subscripts** 

*in* in

*1* stream 1 *2* stream 2 *av* average *b* blade *c* cold stream *gen* generation
