The Entropy Generation Minimisation based on the Revised Entropy Generation Number

Transcription

1 Int. J. Exergy, Vol. 7, No. 5, The Entropy Generation Minimisation based on the Revised Entropy Generation Number Jiangfeng Guo, Lin Cheng and Mingtian Xu* Institute of Thermal Science and Technology, Shandong University, Jinan , China Fax: *Corresponding author Abstract: In the present work, an improved Entropy Generation Minimisation (EGM) approach aiming at minimising the revised entropy generation number which is the non-dimensionalised entropy by the ratio of heat flow to the input temperature of the cold fluid is developed for a plate-fin heat exchanger design with multiple design variables with the help of genetic algorithm. It is found that the approach can decrease the total fan power dramatically and improve the exchanger effectiveness simultaneously. Finally, this approach is applied to a heat recovery system where the heat exchanger works as a component of the system. Keywords: EGN; entropy generation number; plate-fin heat exchanger; GA; genetic algorithm; optimisation design; EGM; entropy generation minimisation; revised entropy generation number; optimisation design. Reference to this paper should be made as follows: Guo, J., Cheng, L. and Xu, M. (2010) The Entropy Generation Minimisation based on the Revised Entropy Generation Number, Int. J. Exergy, Vol. 7, No. 5, pp Biographical notes: Jiangfeng Guo received his BEng in Thermal Energy and Power Engineering from the North China Electric Power University, Baoding, China, in Currently, he is a PhD Research Student at Shandong University, Jinan, China. His research focuses on heat transfer enhancement. Lin Cheng, a Doctor of Engineering, is now a Chief Professor and Director of Institute of Thermal Science and Technology in Shandong University. He is also the Chief Scientist of National Basic Research Program of China and Senior Co-Research Scholar of European Center for Nuclear Research (CERN). He currently holds the Chang Jiang Scholarship from the Chinese Ministry of Education. After Mingtian Xu received BSc in Computational Mathematics and MEng in Solid Mechanics from Shandong University in 1987 and 1990, respectively, he first took up an Assistant Professor position from 1990 to 1994, then Associate Professor from 1995 to 1999, in Shandong University. He obtained Copyright 2010 Inderscience Enterprises Ltd.

2 608 J. Guo et al. his PhD Degree from the University of Hong Kong in After working in Forschungszentrum Dresden-Rossendorf, Germany, for more than four years as a Research Fellow, he joined Shandong University as a Professor in His research interests include magnetohydrodynamics, microscale heat transfer, heat transfer enhancement technology, and multiscale transport. 1 Introduction The last decades witness the rapid development of thermodynamic modelling and analysis of heat exchangers in the framework of the second law of thermodynamics (Bejan, 1977, 1982, 1987, 1995, 1996; Yilmaz et al., 2001). The highlight is the EGM approach developed by Bejan (1977, 1982, 1987, 1995, 1996). In this approach, the entropy generation is employed to measure the thermodynamic irreversibilities caused mainly by heat conduction across the finite temperature difference, fluid friction and fluid mixing, which are detrimental to the heat exchanger performance. On the basis of the concept of entropy generation, several heat exchanger performance criteria were proposed, such as the entropy generation rate (Bejan, 1995), Entropy Generation Number (EGN) defined by dividing the entropy generation rate by the capacity flow rate mc & p (Bejan, 1979, 1982), augmentation EGN (Bejan, 1982), heat exchanger reversibility norm (Sekulic, 1990), Witte Shamsundar efficiency (Witte and Shamsundar, 1983) and local EGN (Sciubba, 1996). These criteria not only have their own characteristics and constraints, but also are interrelated as described in Yilmaz et al. (2001). Among them, the most frequently applied one is the EGN proposed by Bejan (1979, 1982, 1996). With the EGN taken as the objective function, Bejan developed an optimisation design method for counter-flow heat exchangers (Bejan, 1977, 1978), which was applied to a shell and tube regenerative heat exchanger to obtain the minimum heat transfer area when the amount of heat transfer units was fixed. Grazzini and Gori (1988), Sekulic (1990), Zhang et al. (1997), Ordonez and Bejan (2000) and Bejan (2001, 2002) showed that the geometries of heat exchangers can be optimised by the thermodynamic optimisation subject to certain constraint conditions. An analytical method for EGM was also developed to optimise the dimensions of several different fin configurations (Poulikakos and Bejan, 1982). Bejan et al. (1996) demonstrated that EGM may be used by itself in the preliminary stage of design to identify trends and the existence of optimisation opportunities. Vargas et al. (2001) presented an approach to determine the internal geometric configuration of a tube bank by optimising the global performance of the installation that uses the cross-flow heat exchanger. In Oğulatu et al. (2000) based on the second law of thermodynamics, a balanced cross-flow plate-type heat exchanger operating with unmixed fluids was analysed; the variations of the minimum EGN with respect to the flow path length, dimensionless mass velocity, dimensionless heat transfer area and dimensionless heat transfer volume were investigated. Reddy et al. (2002) derived an expression of the EGM for a waste heat recovery steam generator, which consists of an economiser, an evaporator and a superheater, and studied the influences of various non-dimensional operating parameters on the EGN. In Sahiti et al. (2008), the impact of the heat exchanger flow length and the pin length on the EGN was explored; it was found that a larger number of passages with smaller pin height in the given frontal

3 The Entropy Generation Minimisation 609 area of heat exchanger are more preferable than less heat exchanger passages with larger pin height. In Mishra et al. (2009), an optimisation technique based on the GA was developed for the cross-flow plate-fin heat exchanger design with the aim of minimising the commonly used EGN. However, the EGN, which is usually employed to quantify the irreversibilities occurring in the heat exchanger, suffers from the entropy generation paradoxes (Hesselgreaves, 2000). To resolve these paradoxes, a dimensionless EGN was defined by dividing the entropy generation rate by Q/T e (Q is the heat transfer rate and T e is the ambient temperature) (Witte and Shamsundar, 1983; London and Shah, 1983), whereas introducing a new parameter T e into the definition causes further complication (Yilmaz et al., 2001). Recently, a more appropriate means of non-dimensionalising the entropy generation by Q/T c,i (T c,i is the inlet temperature of the cold fluid) was proposed by Hesselgreaves (2000) and the dimensionless EGN defined in such a way was called the REGN and denoted as N s1. To our knowledge, there have been few reports about the application of REGN in the optimisation design of the plate-fin heat exchanger. In this work, the REGN is taken as the objective function, and a multivariable optimisation design problem for the plate-fin heat exchanger is formulated and solved by the GA. Furthermore, this approach is extended to the optimisation design of thermal system in which the heat exchanger works as a component. Given the fact that there is little application of EGM with REGN as the objective function in the thermal system design, our work may arouse the interest in this field. 2 The EGM approach for plate-fin heat exchangers The plate-fin heat exchangers as shown in Figure 1 are widely applied in various industrial applications (Kuppan, 2000). In this section, the EGM approach is employed to the optimisation design of the plate-fin heat exchanger with the REGN as the objective function. In the following discussion, the conventional assumptions for the heat exchanger design are adopted, such as no longitudinal heat conduction, negligible potential and kinetic energy changes, negligible heat transfer between the exchanger and its surroundings (Shah and Sekulic, 2003). Figure 1 Plate-fin heat exchanger

4 610 J. Guo et al. 2.1 Theoretical analysis For a cross-flow plate-fin heat exchanger, the Reynolds number is defined as follows, G d Re = e, (1) µ where G is the fluid mass velocity, d e is the hydraulic diameter of flow passages and µ is the dynamic viscosity. The heat transfer factor j for plate-fin heat exchanger is (Kuppan, 2000): 2/3 j = St Pr (2) where St is Stanton number and Pr is Prandtl number. Kays and London (1984) presented plots of the heat transfer factor j and fanning friction factor f vs. Reynolds number for the tube tank, tube-fin heat exchangers, and other plate-fin surface geometries. Meanwhile, many heat transfer correlations for plate-fin heat exchangers are available in Qian (2002). Among them, the fin performance curves introduced by KOBELCO in Japan are widely used in China, and the reliability of the data has been proved by practice. The curves are shown in Figure 2, and the corresponding correlations for the plate-fin heat exchangers with offset strip fins, when 300 Re 7500, are written as (Qian, 2002), ln j (ln Re) (ln Re) ln Re , = + + (3) 2 ln f = (ln Re) (ln Re) (4) Figure 2 Performance curves of fins with different geometries introduced by KOBELCO in Japan 1: Rectangular fin; 2: Offset strip fin; 3: Perforated fin. Source: Qian (2002) Thus, the heat transfer coefficient in one side of the fluids is expressed as, cg p α = Stc pg = f(re). (5) 2/3 Pr

5 The Entropy Generation Minimisation 611 When the heat capacity rate ratio C* is equal to one, the entropy generation rate for the cross-flow heat exchanger is (Oğulatu et al., 2000), Tho, Tco, Ph Pc S & gen = mc & p ln + mcp ln mrln 1 mrln 1, T & hi, T & ci, P & hi, P (6) ci, where m& is the mass flow rate, T is the temperature, R is the ideal gas constant, P is the pressure, P is the pressure drop, the subscripts h and c refer to the hot and cold fluids, respectively, the subscripts i and o refer to the inlet and outlet of heat exchanger, respectively. The REGN N s1 is defined by dividing S & gen by Q/T c,i, N s1 1 = {ln[1 ε(1/ τ 1)] + ln[1 + ε(1/ τ 1)]} ε(1/ τ 1) R P h R P c ln 1 ln 1, εcp(1/ τ 1) P h, i εcp(1/ τ 1) P c, i where τ = T c,i /T h,i is the ratio of the cold fluid inlet temperature to the hot fluid inlet temperature and ε is the heat exchanger effectiveness. The first line of the right side of equation (7) represents the dimensionless entropy generation N s1t caused by the heat conduction across finite temperature difference; the last two terms stand for the dimensionless entropy generation N s1f caused by the flow friction. The exchanger effectiveness for the single-pass unmixed-unmixed cross-flow heat exchanger with C* = 1 is (Kuppan, 2000), ε = 1 exp{ C * Ntu [exp( C * Ntu ) 1]} (8) where Ntu is the number of exchanger heat transfer units. For the core pressure drop of the plate-fin exchanger, the entrance, momentum and exit effects on pressure drop are neglected in our calculation. Therefore, the pressure drop and the number of exchanger heat transfer units are expressed as follows (Bejan, 1982; Oğulatu et al., 2000) 2 P 4L G = f, (9) P de 2ρP 4L 4L Pr 2/3, Ntu = St = j (10) d d e e where ρ is the fluid density and L is the length of flow passage. Substituting equations (8) (10) into equation (7) gives rise to the final expression of the REGN. This expression describes the dependence of the REGN on various design parameters. Let us consider a gas-to-gas plate-fin heat exchanger. The known parameters are listed in Table 1. Figure 3 shows the variation of the REGN with the ratio of the flow passage length to the hydraulic diameter (L/d e ). From this figure, one can see that the dependences of N s1t and N s1f on L/d e demonstrate the opposite behaviours, therefore there exists an optimal value for L/d e, where the REGN reaches its minimum. This indicates that the appropriate selection of the parameters of the considered heat exchanger can lead to the minimum entropy generation. (7)

6 612 J. Guo et al. Table 1 The known data for gas-to-gas heat exchanger Parameters Values Inlet temperature of hot fluid T h,i ( C) 80 Inlet temperature of cold fluid T c,i ( C) 20 Inlet pressure of two streams P i (MPa) 0.45 Specific heat of two fluids c p (J/(kg K)) Dynamic viscosity of two fluids µ (Pa s) Density for two fluids ρ (kg/m 3 ) Ideal gas constant for air R (J/(kg K)) 287 Mass flow velocity G (kg/(m 2 s)) 20 Figure 3 Variations of the revised EGN with respect to L/d e (Re = 3500) 2.2 Optimisation design of the plate-fin heat exchanger The REGN defined in equation (7) is taken as the objective function, the admissible pressure drop, the requirements from the design standard and users are set to the constraint conditions and some geometrical parameters are selected as the design variables, the optimisation design problem of the plate-fin heat exchanger is thus formulated. One may use the traditional approaches, which require the information of the gradients of objective functions to solve the optimisation problem. Unfortunately, they suffer from getting trapped at the local optimum and cannot ensure that the global optimal solution is achievable (Abu and Munawar, 2007). In comparison with other approaches, the GA demonstrates appealing features in solving optimisation problems. First, it provides a high level of robustness by simulating nature s adaptation in the evolution process (Oh et al., 1999). More importantly, GA has very strong capability to find the global optimum (Fanni et al., 1997). Therefore, the GA (Houck et al., 1995) is employed to seek the global optimal solution of the optimisation problem. The GA is based on the natural selection, which was found in biological evolution processes. In tackling the optimisation problem, before a GA is put to work, a method is needed to encode potential solutions in a form that a computer can process. It regards a potential solution as an individual, and can be represented by a set of parameters, which can be encoded values in binary form as the genes of a chromosome. Initial population of individuals is formed from a random set of solutions, and then next generations are generated through some operators discussed in the following.

7 The Entropy Generation Minimisation 613 A metric called a fitness function allows each potential solution (individual) to be quantitatively evaluated. After a random initial population in the ranges of design variables is generated, the algorithm creates a sequence of new generations iteratively until the stopping criterion is met. In the process, offspring are generated by merging two individuals in current generation with a crossover operator, or by modifying a chromosome with a mutation operator. A new generation is formed by some parents and offspring based on fitness values, the population size is kept constant by eliminating the inferior ones. The chromosomes with higher fitness values have higher probabilities to survive; this guarantees the algorithm converges to a best individual after certain generations, which probably represents the best solution of the given problem (Goldberg, 1989; Holland, 1975). The flow chart of a GA is presented in Figure 4. Figure 4 Flow chart of genetic algorithm To validate the reliability and accuracy of GA used in this work, two typical test examples are considered. The first one is to search the global minimum point of the following function, which is written as follows, 5 2 f( x) xi x i i= 1 =, (11) where x i (i = 1, 2,, 5) are variables. The function f(x) is a continuous unimodal function; it has only one minimum point located at (0, 0, 0, 0, 0), and the minimum value is equal to 0. The initial population is generated randomly within the ranges of parameters, whose size is set to 20 and remain the same in the searching process. The termination condition is that the number of generations is not greater than 500. The global optimal solution obtained by this approach is shown in Table 2. Evidently, the GA has very high accuracy.

8 614 J. Guo et al. Table 2 The global optimal solution of the f(x) function obtained by genetic algorithm x 1 x 2 x 3 x 4 x 5 f(x) For the second problem, we consider Shaffer s F 6 function defined as follows: ( x y ) sin f( x, y) = 0.5, 100 < x, y < ( ( x + y )) (12) The function has countless local maximum values, but it has only one global maximum value located at (0, 0) as shown in Figure 5. The global maximum value is surrounded by a circle of local maximum values, which are all equal to Hence, when searching the global maximum point, it is quite easy to be trapped at the local maximum points. The GA is adopted to solve this function, the size of population is set to 20, and the maximum number of generations is set to 500. The searching process and the final result are documented in Table 3. One can see that the GA has not got trapped at local optimum points and eventually gives rise to the exact global optimal solution. This example shows that the GA is a reliable and powerful tool to tackling the global optimisation problems and usually has high accuracy. Figure 5 The three-dimensional image of Shaffer s F 6 function (see online version for colours) Table 3 Shaffer s F 6 function solving process with genetic algorithm Generations x y f(x, y)

9 The Entropy Generation Minimisation Example 1 In this example, a cross-flow plate type heat exchanger with offset strip fin is considered. In this heat exchanger, the hot and cold fluids are balanced, namely C* = 1. The known data for the plate-fin heat exchanger under consideration have been documented in Table 1. The ranges of the design variables and the constraint conditions are given in Table 4 according to the national standards of China as well as the characters of the heat exchanger (Qian, 2002). On the basis of the data in Tables 1 and 4, the other parameters can be determined. The hydraulic diameter is calculated as follows d e = 2xy x + y (13) where x is the internal span of fins, y = H δ and it is the inside height of fins, δ is the thickness of fin and is set to 0.3 mm in the following calculation. The effective cross-section area in one layer is given as L fi = xy. (14) s Table 4 Design variables as well as their ranges and constraint conditions for Example 1 Variables Ranges Design variables The fin height H (mm) The fin pitch x (mm) The number of hot side layers n The length of PFHE core L(m) Constraint conditions Pressure drop P h in hot side (Pa) <5000 Pressure drop P c in cold side (Pa) <6000 Re in two sides 300 < Re < 7500 In this example, both fluids are assumed as ideal gases and their mass flow rates are the same, so the effective width is equivalent to the length of flow channel. Then, heat transfer area of one layer can be obtained by F x y L s 2 i = 2( + ) /. (15) The mass flow velocity in the channel reads m G = &. (16) nf i Since the thickness of fin is usually far less than the fin offset length, the transverse heat conduction across the fin is thus neglected, and the staggered arrangement for hot and cold streams is assumed, then the fin efficiency can be calculated as follows (Kays and London, 1984).

10 616 J. Guo et al. 2α H tan λδ w 2 η f = (17) 2α H λδ 2 w where λ w is the thermal conductivity of fins and equals to 190 W/(m K), and H is the height of fin. So, the total fin effective efficiency is y ηo = 1 (1 η f ). x+ y (18) If the wall and fouling thermal resistances are negligible, the total heat transfer coefficient based on the hot fluid side is K h 1 1 F h = + αη h oh αη c oc Fc 1 where F h and F c denote the heat transfer areas in the hot and cold fluid sides, respectively. The number of heat transfer units based on the hot fluid side is expressed as KhFn i Ntu =. (20) mc & p The heat transfer rate can be calculated as follows Q = mc & ε ( T T ) (21) p hi, ci, where the exchanger effectiveness ε is expressed by equation (8). The total power of fans is written as (Caputo et al., 2008) (19) 1 m& h m& c W = Ph + Pc η ρh ρc (22) where η is the fan efficiency. Furthermore, the following JF factor is defined for evaluating the heat exchanger performance (Yun and Lee, 2000): j/ j JF = ( f / f ) (23) R 1/3 R where j R and f R are the reference values of j and f factors, respectively. On the basis of the known data of the considered heat exchanger and the above-mentioned heat transfer calculation, we attempt to optimise the design of the plate-fin heat exchanger. The objective function is given by equation (7), the numbers of the initial population and maximum generation are set to 20 and 500, respectively. The whole process of searching the optimum solution of the optimisation design problem by the generic algorithm is described as follows:

11 The Entropy Generation Minimisation 617 Step 1: Choose the fitness function. In the following calculation, the minus REGN is chosen as the fitness function. Step 2: Determine the design variables and the ranges of their values as well as the constraint conditions. The selected design variables and the ranges of their values as well as the constraint conditions for Example 1 are listed in Table 4. Step 3: Once the values of design variables are given, the other parameters of the heat exchanger can be calculated by the traditional heat exchanger design process and the formulas given in the above-mentioned discussion as well as the known information of the heat exchanger listed in Table 1. Step 4: Calculate the value of fitness function for every individual (potential solution), and judge whether it exceeds the acceptable constraints, if so the fitness of this individual is set to a very small value. After that, check if the termination condition is met. If so, the individual with the maximum value of the fitness function is the optimum solution of the optimisation design problem of the plate-fin heat exchanger, otherwise, move to the next step. Step 5: Choose the parents among all the individuals based on their values of fitness function calculated in step 4 and generate next generation through cross operator and mutation operator. Then, return to step 4. For the details related to the GA, please refer to Houck et al. (1995), Goldberg (1989) and Holland (1975). On the basis of the GA toolbox in Matlab (Houck et al., 1995), a program is designed to solve the optimisation design problem of the plate-fin heat exchanger. After 500 generations of evolution, the process is terminated and the best individual (optimum solution) is obtained. For illustrating the evolution process, the variation of the REGN of the best individual in every generation with respect to the number of generations is displayed in Figure 6. Figure 6 The REGN vs. the number of generations From Figure 6, one can see that the REGN decreases with increasing the number of generations. When the number of generations exceeds 100, the REGN tends to be stable. This indicates that the GA is convergent. The parameters of the initial design plan and the optimum result are listed in Table 5. From this table, one can see that in comparison with the randomly generated initial design of the heat exchanger, the optimal design solution reduces the power of fans by 63.3% and increases the exchanger effectiveness by 4.13%. This is achieved by

12 618 J. Guo et al. appropriately selecting the fin height, the number of layers, the flow passage length and the fin pitch. Table 5 The initial parameters and optimum result for the plate-fin heat exchanger H (mm) X (mm) n L (m) d e (mm) Re Ntu ε W (kw) N s1h N s1f N s1 Initial Final The variations of the exchanger effectiveness, the pressure drop, the number of the heat transfer units and JF factor of the best individual for every generation during the evolution process are depicted in Figure 7. From Figure 7(a), one can see that with decreasing the REGN, the exchanger effectiveness tends to increase. Figure 7(b) illustrates that the significant decrease in the power of fans occurs with the decline of the REGN. From Figure 7(c), one can see that when the iterative process proceeds, the overall tendency of the number of heat transfer units is to increase, but in the final stage, the number of heat transfer units has slight change. From Figure 7(d), one can see that the JF factor is larger than 1, and becomes larger and larger with decreasing N s1. Therefore, the heat exchanger performance becomes better and better (Yun and Lee, 2000), and reaches the best when N s1 is minimum. Figure 7 Variations of the exchanger effectiveness, power of fans, number of heat transfer units and JF factor during the iterative process of genetic algorithm: (a) the exchanger effectiveness vs. N s1 ; (b) the power of fans vs. N s1 ; (c) variation of Ntu with the number of generations and (d) JF factor vs. N s1 (a) (b) (c) (d)

13 The Entropy Generation Minimisation 619 Note that although the initial individuals are generated randomly, they satisfy the heat load requirement, the constraint conditions and the design standards, therefore they are eligible heat exchanger design plans. Furthermore, it is found that for different initial individuals the tendency that the initial design goes to the optimum design through the optimisation process is similar Example 2 In this example, we consider the optimisation problem of a cross-flow plate-fin heat exchanger with a given heat load. In this heat exchanger, the hot and cold fluids are oxygen and air, respectively. The information of the heat exchanger under consideration is documented in Table 6 (the symbol represents the unknown data). The design variables and their ranges of values are listed in Table 7. Note that the heat capacity flow rates of the hot and cold fluids are not same in this example. Table 6 The known data for the considered heat exchanger with the fixed heat load in Example 2 Parameters Hot gas (oxygen) Cold gas (air) Inlet temperature T i ( C) Outlet temperature T o ( C) 30 Mass flow rate m& (kg/s) Density ρ (kg/m 3 ) Specific heat at constant pressure c p (J/(kg K)) Dynamic viscosity µ (Pa s) Inlet pressure P (MPa) Thermal conductivity λ (W/(m K)) Table 7 Design variables as well as their ranges and constraint conditions for Example 2 Variables Ranges Design variables The fin height in hot side H h (mm) The fin height in cold side H c (mm) The fin pitch in hot side x h (mm) The fin pitch in cold side x c (mm) The number of hot side layers n The effective width B (m) Constraint conditions Pressure drop P h in hot side (Pa) <5000 Pressure drop P c in cold side (Pa) <6000 Re in two sides 300< Re < 7500

14 620 J. Guo et al. For the effectiveness of the cross-flow heat exchanger with both fluids unmixed, although its series expression is available (Kuppan, 2000), it is not convenient to use, therefore in our treatment, an approximate formula is developed by curve fitting based on the data presented in Qian (2002). ε = Ntu C C Ntu C * C * * * 2. (24) The heat load is Q = ( mc & ) ( T T ). (25) p h h, i h, o The flow passage length is expressed as Q L = K t 2 n( x + y ) B s h lm h h h (26) where t lm is the log-mean temperature difference. For the convenience of comparison, all parameters are calculated based on the hot fluid side. Then, the similar optimisation process as used in Example 1 is applied. The variations of the REGN, Ntu and W with the number of generations in the optimisation process are shown in Figure 8. From Table 6, one can see that the heat load and the mass flow rate are given in this example. Thus, the exchanger effectiveness is fixed, which can also be seen from the N s1t -curve in Figure 8(a) and Ntu-curve in Figure 8(b). However, the entropy generation caused by the fluid friction experiences a sharp decrease during the iterative process as shown in Figure 8(a). The similar phenomenon occurs for the total fan power as shown in Figure 8(b); the total fan power decreases by 78% through the optimisation process, and the JF factor defined by equation (23) increases from 1 to 1.5. The comparison between one initial and the optimal parameters is shown in Table 8. From Table 8, one can see that the effectiveness and heat transfer EGN fix at 0.6 and , respectively, while the power of fans and the EGN caused by fluid friction fall to 36 kw and from 165 kw and , respectively. Therefore, the comprehensive performance of heat exchangers is significantly improved. Figure 8 Variations of the REGN, Ntu and W with respect to the number of generations: (a) the revised EGN vs. the number of generations and (b) Ntu and W vs. the number of generations (a) (b)

15 The Entropy Generation Minimisation 621 Table 8 The initial parameters and optimum result for the plate-fin heat exchanger for Example 2 H h (mm) H c (mm) x h (mm) x c (mm) n B (mm) Ntu ε W (kw) N s1h N s1f N s1 Initial Final Example 3 In this subsection, the irreversibility minimisation at the system level is investigated. More specifically, in this example we consider the optimisation design of a waste heat recovery ventilation system in which the heat exchanger functions as a component. It is assumed that the room temperature is maintained at 20 C and the outdoor temperature is 5 C. Therefore, the outdoor air needs to be heated from 5 C to 20 C before getting into the room. The known data about the system is listed in Table 9. If the cold air is only heated by electric heater, the power consumption can be calculated as follows: C = ( mc & ) ( T T ) t (27) H p c r c, o where T r is the room temperature; t is the operating hours in one year and is assumed to 4000 h/yr in our calculation. So, the annual power consumption C H is kwh. To recycle the heat of exhausted air from the room, it first goes into a counter-flow plate-fin heat exchanger where the hot air and the cold air can exchange heat before being discharged from room. Such a waste heat recovery ventilation system is depicted in Figure 9. The information of the system is listed in Table 9. On the basis of this known information, the annual electrical energy consumptions for heating the cold air by electric heater and for driving fans are equal to 2492 kwh and kwh, respectively, therefore the reduction of annual power consumption by using the counter-flow plate-fin heat exchanger to recover the waste heat is 7544 kwh. Table 9 The known data for the considered waste heat recovery ventilation system in Example 3 Parameters Hot air Cold air Inlet temperature T i ( C) 20 5 Mass flow rate m& (kg/s) Density ρ (kg/m 3 ) Specific heat at constant pressure c p (J/(kg K)) Dynamic viscosity µ (Pa s) Inlet pressure P (Pa) Thermal conductivity λ (W/(m K))

16 622 J. Guo et al. Figure 9 Waste heat recovery ventilation system To save more energy, the system needs to be optimised. It is assumed that the heat exchange area is fixed at 20 m 2, and the power of fans cannot exceed 30 W, the entropy generation caused by the fluid mixing is neglected, and the heat transfer between heat exchanger and environment is negligible. Then, the total entropy generation rate of the waste heat recovery ventilation system is: S& = S& + S& + S& + S& gen gen, H gen, T gen, F gen, E T T T = ( mc & ) ln + ( mc & ) ln + ( mc & ) ln r ho, co, p c p h p c T T T co, hi, ci, P P ( T T ) h c ho, e ln 1 ln 1 ( ) h mr & c mc & p h. P P T hi, ci, e mr & + The first term in equation (28) is the rate of entropy generation due to cold air heating by electric heater, the last term accounts for the rate of entropy generation due to heat transfer from hot air leaving the exchanger for the environment at temperature T e, and the other terms account for the rate of entropy generation caused by the heat conduction across finite temperature difference and fluid friction in the heat exchanger. The dimensionless EGN can be obtained by dividing equation (28) by Q/T c,o and it is taken as the objective function in the optimisation process. In this example, only three deign variables are considered, namely the fin height H, the internal span of fins x, the layer number n, and their ranges of values are mm, mm and 4 30, respectively. The similar optimisation process as the last two examples is implemented. The optimum solution is thus obtained and its information is listed in Table 10. (28) Table 10 The initial parameters and optimum result for the waste heat recovery ventilation system H (m) x (m) n K h (W/m 2 K) Q (kw) T h,o (K) T c,o (K) W (W) S & g, H (W/K) S & g, T (W/K) S & g, P S & g, E (W/K) (W/K) N s1 Initial Final From Table 10, it is evident that S & gen,h and S & gen,e are dominant in the total entropy generation, and decrease by 44.7% and 44.3% in comparison with the initial design, respectively. Besides, the heat transfer rate increases from kw to kw, and the fan power increases from 3.37 W to 29.6 W through the optimisation process.

17 The Entropy Generation Minimisation 623 Note that the irreversibility S & gen,f caused by fluid friction is quite small in comparison with other irreversibility. The amount of power saving through the optimisation process can be calculated by subtracting the increase in fan power from the increase in heat transfer rate, which is equal to kw. Then, the optimisation design of the waste heat recovery ventilation system can lead to 1107 kwh energy saving per year. The capital investment has not changed, since the heat transfer area is kept constant. Therefore, a notable effect is achieved by the optimisation design of the waste heat recovery ventilation system. 3 Concluding remarks The REGN is more advantageous than the EGN in the sense that it avoids the entropy generation paradoxes. Therefore, we take the REGN as the objective function and develop an optimisation design scheme for the plate-fin type heat exchanger. It is found that the optimisation process can improve the heat exchanger performance significantly in the case that the heat load is kept constant; the dramatic decrease in the power of fans is achieved. Finally, the application of this EGM in a waste heat recovery ventilation system leads to a notable energy saving. Acknowledgement The support of our research by National Basic Research Program of China (Project No. 2007CB206900) is greatly appreciated. References Abu, B.V. and Munawar, S.A. (2007) Differential evolution strategies for optimal design of shell-and-tube heat exchangers, Chemical Engineering Science, Vol. 62, pp Bejan, A. (1977) The concept of irreversibility in heat exchanger design: counter-flow heat exchangers for gas-to-gas applications, ASME Journal of Heat Transfer, Vol. 99, pp Bejan, A. (1978) General criteria for rating heat exchanger performance, International Journal of Mass and Heat Transfer, Vol. 21, No. 5, pp Bejan, A. (1979) A study of entropy generation in fundamental convective heat transfer, ASME Journal of Heat Transfer, Vol. 101, No. 4, pp Bejan, A. (1982) Entropy Generation through Heat and Fluid Flow, Wiley, New York. Bejan, A. (1987) The thermodynamic design of heat and mass transfer processes and devices, International Journal of Heat and Fluid Flow, Vol. 8, No. 4, pp Bejan, A. (1995) Entropy Generation Minimization, CRC Press, New York. Bejan, A. (1996) Entropy generation minimization: the new thermodynamics of finite-size devices and finite-time processes, Journal of Applied Physics, Vol. 79, No. 3, pp Bejan, A. (2001) Thermodynamic optimization of geometry in engineering flow systems, Exergy, An International Journal, Vol. 1, No. 4, pp Bejan, A. (2002) Fundamentals of exergy analysis, entropy generation minimization, and the generation of flow architecture, International Journal of Energy Research, Vol. 26, No. 7, pp

18 624 J. Guo et al. Caputo, A.C., Pelagagge, P.M. and Salini, P. (2008) Heat exchanger design based on economic optimization, Applied Thermal Engineering, Vol. 28, No. 10, pp Fanni, A., Marchesi, M., Serri, A. and Usai, M. (1997) A greedy genetic algorithm for continuous variables electromagnetic optimization problems, IEEE Trans. on Mag., Vol. 33, No. 2, pp Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, Mass. Grazzini, G. and Gori, F. (1988) Entropy parameters for heat exchanger design, International Journal of Heat and Mass Transfer, Vol. 31, No. 12, pp Hesselgreaves, J.E. (2000) Rationalisation of second law analysis of heat exchanger, International Journal of Heat Mass Transfer, Vol. 43, No. 22, pp Holland, J.H. (1975) Adaptation in Nature and Artificial System, The University of Michigan Press, Ann Arbor, USA. Houck, C.R., Joines, J.A. and Kay, M.G. (1995) A Genetic Algorithm for Function Optimization: A Matlab Implementation, Technical Report NCSU-IE-TR-95-09, North Carolina State University, Raleigh, NC. Kays, W.M. and London, A.L. (1984) Compact Heat Exchanger, 3rd ed., McGraw-Hill, New York. Kuppan, T. (2000) Heat Exchanger Design Handbook, Marcel Dekker Inc., New York. London, A.L. and Shah, R.K. (1983) Costs of irreversibilities in heat exchanger design, Heat Transfer Engineering, Vol. 4, pp Mishra, M., Das, P.K. and Sarangi, S. (2009) Second law based optimisation of crossflow plate-fin heat exchanger design using genetic algorithm, Applied Thermal Engineering, Vol. 29, pp Oğulatu, R., Doba, F. and Yilmaz, T. (2000) Irreversibility analysis of cross flow heat exchangers, Energy Conversion and Management, Vol. 41, No. 15, pp Oh, Y.H., Kim, T. and Jung, H.K. (1999) Optimal design of electric machine using genetic algorithm coupled with direct method, IEEE Trans. on Mag., Vol. 35, No. 3, pp Ordonez, J.C. and Bejan, A. (2000) Entropy generation minimization in parallel-plates counterflow heat exchangers, International Journal of Energy Research, Vol. 24, No. 10, pp Poulikakos, D. and Bejan, A. (1982) Fin geometry for minimum entropy generation in forced convection, ASME Journal of Heat Transfer, Vol. 104, No. 4, pp Qian, S.W. (2002) Heat Exchanger Design Handbook, Chemical Industrial Press, Beijing (in Chinese). Reddy, B.V., Ramkiran, G., Kumar, K.A. and Nag, P.K. (2002) Second law analysis of a waste heat recovery steam generator, International Journal of Heat and Mass Transfer, Vol. 45, No. 9, pp Sahiti, N., Krasniqi, F., Fejzullahu, X.H., Bunjaku, J. and Muriqi, A. (2008) Entropy generation minimization of a double-pipe pin fin heat exchanger, Applied Thermal Engineering, Vol. 28, Nos , pp Sciubba, E. (1996) A minimum entropy generation procedure for the discrete pseudo-optimization of finned-tube heat exchangers, Rev. Gen. Therm., Vol. 35, No. 416, pp Sekulic, D.P. (1990) The second law quality of energy transformation in a heat exchanger, ASME Journal of Heat Transfer, Vol. 112, No. 2, pp Shah, R.K. and Sekulic, D.P. (2003) Fundamentals of Heat Exchanger Design, John Willey, Hoboken. Vargas, J.V.C. Bejan, A. and Siems, D.L. (2001) Integrative thermodynamic optimization of the crossflow heat exchanger for an aircraft environmental control system, ASME Journal of Heat Transfer, Vol. 123, No. 4, pp Witte, L.C. and Shamsundar, N. (1983) A thermodynamic efficiency concept for heat exchange devices, Journal of Engineering for Power-Transactions of the ASME, Vol. 105, pp

19 The Entropy Generation Minimisation 625 Yilmaz, M., Sara, O.N. and Karsli, S. (2001) Performance evaluation criteria for heat exchangers based on second law analysis, Exergy, An International Journal, Vol. 1, No. 4, pp Yun, J.Y. and Lee, K.S. (2000) Influence of design parameters on the heat transfer and flow friction characteristics of the heat exchanger with slit fins, International Journal of Heat and Mass Transfer, Vol. 43, No. 14, pp Zhang, L.W., Balachandar, S., Tafti, D.K. and Najjar, F.M. (1997) Heat transfer enhancement mechanisms in in-line and staggered parallel-plate fin heat exchangers, International Journal of Heat and Mass Transfer, Vol. 40, No. 10, pp Nomenclature B Effective width (m) c p Specific heat at constant pressure (J/(kg K)) C* Heat capacity flow rate ratio d e Hydraulic diameter (m) f Friction factor f i Effective cross section area for one Layer (m 2 ) F Heat transfer area (m 2 ) F i Heat transfer area for one layer (m 2 ) G Fluid mass velocity (kg/(s m 2 )) H The height of fin (m) j Heat transfer factor JF JF factor K Total heat transfer coefficient (W/(m 2 K)) L Length of flow passage (m) m& Mass flow rate (kg/s) N Number of layers N s Entropy Generation Number (EGN) N s1 Revised Entropy Generation Number (REGN) Ntu Number of heat transfer units P Pressure (Pa) Pr Prandtl number Q Heat transfer rate (W) R Ideal gas constant (J/(kg K)) Re Reynolds number St Stanton number S & Entropy generation rate (W/K) gen t Yearly running time (h) T Temperature (K) T e Environment temperature (K) W Power of fans (W)

20 626 J. Guo et al. x Internal span of fins (m) y Inside height of fins (m) Greek symbols P Pressure drop (Pa) t lm δ η η f η o λ Log-mean temperature difference (K) The thickness of fin (m) Fan efficiency Fin efficiency Total fin efficiency Thermal conductivity (W/(m K)) µ Dynamic viscosity (Pa s) ρ Fluid density (kg/m 3 ) Subscripts c Cold fluid f Fluid h Hot fluid i Inlet o Outlet r Room R Reference w Wall