Optimal design analysis of a tubular heat exchanger network with extended surfaces using multi-objective constructal optimization

Abstract

The present work aims to investigate the influence of extended surfaces (fins) on the multi-objective optimization of a tubular heat exchanger network (THEN). An increase in the heat transfer area using various extended surfaces (fins) to enhance the performance of the heat exchanger was used while considering the effectiveness and total heat transfer area as two objective functions. In addition to the simulation of simple fins, a new set of fins, called constructal fins, was designed based on the constructal theory. Tubular heat exchanger network design parameters were chosen as optimization variables, and optimization results were achieved in such a way as to enhance the effectiveness and decrease the total heat transfer area. The results show the importance of constructal fins in improving the objective functions of heat exchangers. For instance, the simple fins case enhances the effectiveness by up to 5.3% compared to that without fins (usual heat exchanger) while using constructal fins, in addition to the 7% increment of effectiveness, reduces the total heat transfer area by 9.47%. In order to optimize the heat exchanger, the heat transfer rate and cold fluid temperature must increase, and at the same time, the hot exiting fluid temperature should decrease at the same constant total heat transfer area, which is higher in the constructal fins case. Finally, optimized design variables were studied for different cases, and the effects of various fins were reported.

Graphical abstract

Keywords

constructal theory / extended surface / effectiveness / total heat transfer area / multi-objective optimization

1 Introduction

Heat exchangers transfer heat from hot fluids to cold fluids by means of the temperature differences between the two media. They have been frequently used in various industrial applications, including power plants, air conditioning, and the chemical industry. The widespread use of heat transfer in various industries reveals the important role of heat exchangers and has led to the recognition of thermal engineering with these components. Among the different types of thermal equipment, pipe heat exchangers play a more prominent role in industrial applications than other types and have a special share in industries. The reasonable price and large size of heat exchangers, especially in industrial applications and generally high capacities, can be considered as the reasons for the introduction of optimization in thermal engineering. In the meantime, the main task of heat exchangers should not be neglected, and increasing the rate of heat transfer and the growth of thermal efficiency, the optimal design has become important [1,2]. In addition, various tubular heat transfer increases are commercially accessible for application to the shell-and-tube types of heat exchangers favored by the chemical processing industries [3,4].

Many works were conducted for optimization of various kinds of heat exchangers with different objectives and judgment parameters using different procedures. For instance, Hajabdollahi et al. carried out the optimization of various types of heat exchangers involving the shell and tube [5,6], condenser [7], plate fin [8,9], fin tube [10,11], rotary regenerator [12,13], and plate heat exchanger [14] by employing various algorithms including the genetic algorithm [14], particle swarm algorithm [7], firefly algorithm [6], and various objective functions involving total annual cost, effectiveness, pressure drop, exergy efficiency, entropy production, and temperature methodology.

Improving the performance of heat exchangers is one of the most important issues that has always been discussed and has attracted the attention of many researchers. The term comprehensive surface is usually applied to illustrate a significant singular case, including conductive heat transfer within a solid and convective heat transfer from the boundaries of the solid [15]. Bejan presented a theory that introduced a new approach that could be extended to a variety of sciences. This nature-inspired method, known as the constructal theory, states that in order to maintain a finite flow system, changes must be made in such a technique as to create the easiest path for the flow to run. This method focuses on the arrangement of the system and makes it possible to improve the parameters through the system geometry [16–22]. After presenting the theory, new methods derived from the concept of the above theory have been used in various studies. In general, structural optimization can be divided into three categories, i.e., size, shape, and topology optimization [23].

Hosseinnezhad et al. [24] numerically studied the turbulent flow of a water/Al₂O₃ nanofluid in a tubular heat exchanger with two twisted-tape inserts. They stated that by reducing the twist ratio, the counter-swirl flow twisted-tape and the augmentation of the volume fraction of Al₂O₃ nanoparticles in the base fluid enhanced the average Nusselt number. Hajabdollahi et al. [25] investigated the influence of nanoparticle shapes, including blades, cylinders, and bricks, on the thermoeconomic optimization of tubular heat exchanger (HE) networks. The optimal results indicated that the best-studied nanoparticle shape was a blade, followed by brick and cylinder shapes. Poongavanam and Ramalingam [26] conducted an experimental study on the heat transfer and pressure drop characteristics and determined the figure of merit in a tubular heat exchanger. The results illustrated that the Nusselt number, friction factor, and figure of merit improved as a result of shot peening. Skullong et al. [27] suggested a delta-wing tape vortex generator (DWT) for heat transfer augmentation. Their findings indicated the DWT offered a much better efficiency than the plain tube alone. They also obtained optimum DWT parameters. Chingtuaythong et al. [28] suggested a novel V-shaped ring turbulator to enhance the heat transfer. They experimentally evaluated the effect of the V-ring parameters on the efficiency and proposed optimum V-rings. Sadeghi et al. [29] numerically investigated the influence of charging and discharging of multiple layers of phase change materials (PCMs) in coaxial cylinders with a time-periodic boundary condition. Tamna et al. [30] experimentally studied the heat transfer improvement in a round tube by inserting double twisted tapes in common with 30° V-shaped ribs. They found that the heat transfer and pressure drop in terms of the respective Nusselt number and friction factor for the V-ribbed twisted tapes showed an increasing trend with an increase in Re and blockage ratio (BR). Kerche et al. [31] evaluated the microparticulation of whey proteins at low concentrations in a pilot plant tubular heat exchanger (THE). Their findings showed that the variation in flow velocity in the holding section had an insignificant effect on the denaturation of β-Lg and particle size of agglomerates.

3D turbulent flow numerical simulations were performed to investigate the heat transfer and flow friction characteristics of louver fin-and-tube heat exchangers by Sadeghianjahromi et al. [32]. They studied the influences of louver angle, fin pitch, transversal tube pitch, and longitudinal tube pitch on Colburn and friction factors in detail. Muley and Manglik [33,34] experimentally improved the thermal-hydraulic performance of single-phase flows in a plate heat exchanger with mixed chevron plates. They discussed the relative effect of using a mixed chevron plate arrangement and evaluated their improved performance in comparison with parallel-plate channels. Beigzadeh and Eiamsa-Ard [35] developed and modeled a heat transfer improvement system involving a CuO/water nanofluid in a corrugated tube prepared with twisted tape using two well-known artificial neural network techniques. The main objective of this model was to assess the Nusselt number (Nu) and friction factor (f) in the considered heat exchanger.

For the design and optimization of shell and tube heat exchanger (STHE), a new methodology based on the constructal theory was studied by Azad and Amidpour [36]. The results showed a more than 50% decrease in the cost of the heat exchanger. In addition, the constructal theory for minimizing the global pressure drop in a comb-like channel network with self-heating and self-cooling was employed by Lee et al. [37]. Their results indicated that the ideal internal and external aspect ratios of the flow architecture, such that the total pressure drop was minimal. A triangular cooling channel with internal heat generation based on the constructal theory was optimized by Meyer et al. [38]. Hafizan et al. [39] studied the optimum design of heat exchanger networks (HENs) around trading off technical design necessities by considering the economy, such as the capital cost of heat exchangers and utilities. In another study, Yang et al. [40] optimized the STHE by applying a general design methodology that was carried out by constructal theory. Norouzi and Amidpour [41] studied and optimized the steam generator capital cost by changing its volume, and stated that the optimum volume was enhanced when the temperature of the inlet and MFR of the flue gas was increased. Mirzaei et al. [42] improved the effectiveness of an STHE by 28% after constructal optimization. The entire cylindrical pin-fin heat sinks (PFHS) using constructal optimization was performed by Yang et al. [43]. The results provided theoretical support for the optimization design of cylindrical PFHSs. The development of the constructal theory about engineering problems in China over the past decade was investigated by Chen et al. [44]. The constructal design progress for eight types of heat sinks was performed by Chen et al. [45]. The results showed that the optimal constructs were mostly different based on different optimization objectives under the same boundary conditions. The constructal design of disc-shaped heat exchanger (DSHE) and STHE was investigated by Feng et al. [46,47]. Other authors focused on the constructal design of heat exchangers and multi-objective optimization using NSGA-II [48–54]. Moreover, some other studies were conducted concerning the constructal design of H- and X-shaped heat exchangers [55], evaporator [56], supercharged boiler evaporator [57], shell-and-tube condenser [58], supercharged boiler superheater [59], and boiler economizer [60]. The thermo-economic and design calculations of a flat-plate solar collector were investigated by Ganjehkaviri and Mohd Jaafar [61]. Their results also demonstrated that optimum thermodynamic point of constructal FPC a 5.25% increment was obtained as compared with the optimum thermodynamic point of conventional FPC. Improvement in the thermal performance (higher than 0.747) of plate fin heat exchanger (PFHE) was obtained by Hajabdollahi [62] using constructal optimization. Morphing the design to go with the times was investigated by Bejan and Gucluer [63]. It was found that, if the structure was free to morph to maintain its optimal spacing in steps with its changing environment, the time-integrated performance of the morphing object was maximum.

Ramjar et al. [64] optimized the heat transfer of forced convection in a wavy channel with different phase shifts between the upper and lower wavy walls. To optimize the process, they employed a recent procedure called the artificial bee colony (ABC) algorithm and compared it with two other meta-heuristic algorithms, called particle swarm optimization (PSO) and differential evolution (DE). Gürbüz et al. [65] utilized Al₂O₃ and CuO nanoparticles to prepare Al₂O₃–CuO/water hybrid nanofluids. Their results confirmed that applying Al₂O₃–CuO/water hybrid-type nanofluid in a plate-type heat exchanger improved the thermal performance remarkably in comparison with single-type nanofluids and enhanced the thermal efficiency in all plate heat exchangers with various numbers of plates.

In the present study, a new method motivated by the constructal theory was examined to determine the optimum design of a tubular heat exchanger network (THEN). For this goal, both the effectiveness of the heat exchanger and the total heat transfer area were selected as simultaneous objective functions. Then, multi-objective optimization was employed to determine the optimal values of the decision variables and objective functions, which were considered to be geometries of the fin (length and thickness of fin first part, length and thickness of fin second part, and fin number) and heat exchanger configuration (tube diameter, tube length, tube number, tube arrangement, tube pitch ratio, and number of heat exchangers). To obtain a good insight, optimization was performed without fin (usual THEN), and the results were evaluated with a simple fin and constructal fin.

2 Thermal modelling

In the present study, the effectiveness and number of transfer units or (ε-NTU) method was applied for thermal examination of THEN (Fig.1). Fig.2(a) shows a schematic of the cylinder and fin array. Fig.2(b) illustrates the cylinder with the constructal fins array. Increasing the number of branches increases the fulling formation, maintenance, and pressure drop. As a result, in this study, only the two branches or the first-order constructal theory were used.

The subsequent assumptions for modelling are that streams are fully counter-flow; the heat loss from the outside of the shell is negligible; thermal radiation is negligible; the conduction along the fins is one-dimensional; and the conditions are in steady-state.

The heat transfer between the cold and hot fluids is obtained based on

(1)Q=U_{\mathrm{O}}{A}_{\mathrm{o}}{\mathrm{\Delta }T}_{\mathrm{L}\mathrm{M}\mathrm{T}\mathrm{D}},

where {\mathrm{\Delta }T}_{\mathrm{L}\mathrm{M}\mathrm{T}\mathrm{D}} is the logarithmic mean temperature difference which is determined as

(2){\mathrm{\Delta }T}_{\mathrm{L}\mathrm{M}\mathrm{T}\mathrm{D}}=\frac{{\mathrm{\Delta }T}_1-{\mathrm{\Delta }T}_2}{\mathrm{l}\mathrm{n}{\displaystyle \frac{{\mathrm{\Delta }T}_1}{{\mathrm{\Delta }T}_2}}}.

The heat transfer rate is defined by Eq. (3), which considers the first law of thermodynamics,

(3)Q={\left(\dot{m}{c}_{\mathrm{p}}\mathrm{\Delta }T\right)}_{\mathrm{h}}={\left(\dot{m}{c}_{\mathrm{p}}\mathrm{\Delta }T\right)}_{\mathrm{c}},

where the subscripts h and c represent the hot and cold side streams, respectively. In addition, \dot{m} and c_{\mathrm{p}} are the mass flow rates and capacity of a specific thermal, respectively. Furthermore, A_{\mathrm{o}} and U_{\mathrm{O}} are the surface of heat transfer and coefficient of total heat transfer, defined as

(4)U_{\mathrm{O}}=\frac{1}{{\displaystyle \frac{1}{{\eta }_{\mathrm{o}}{h}_{\mathrm{s}}}}+{\left(1/{\eta }_{\mathrm{o}}\right)R}_{\mathrm{f},\mathrm{o}}+{\displaystyle \frac{d_{\mathrm{o}}\mathrm{l}\mathrm{n}(d_{\mathrm{o}}/d_{\mathrm{i}})}{2k_{\mathrm{w}}}}+{\displaystyle \frac{A}{A_{\mathrm{o}}}}{R}_{\mathrm{f},\mathrm{i}}+{\displaystyle \frac{A}{A_{\mathrm{o}}}}{\displaystyle \frac{1}{h_{\mathrm{t}}}}},

(5)A_{\mathrm{o}}=\left(\text{π}l{d}_{\text{o}}{N}_{\text{t}}\right)N_{\text{p}},

where d_{\mathrm{o}}, d_{\mathrm{i}}, k_{\mathrm{w}}, R_{\mathrm{f}}, N_{\text{t}}, N_{\text{p}} and l are the outside tube diameter, inner tube diameter, coefficient of wall thermal conductivity, resistance of overall fouling, number of tubes, tube pass number, and tube length in each pass, respectively. In addition, η_o and h are the efficiency of the fin and coefficient of convective heat transfer, respectively. The subscript s indicates the shell and subscript t denotes the tube.

2.1 Heat transfer coefficient and friction factor

The coefficient of convective heat transfer and friction factor on each side is estimated as [42]

(6)\begin{array}{rl}& h=\left(3.657+\frac{0.0657{\times \left(Re\mathit{P}\mathit{r}\times d_{\mathrm{i}}/l\right)}^{1.33}}{1+0.1Pr{\left(Re\times d_{\mathrm{i}}/l\right)}^{0.3}}\right)\frac{k}{d_{\mathrm{i}}},Re\le \text{2300},\\ & f=\frac{16}{Re},\end{array}

(7)

(8)\begin{array}{rl}& h=\left(\frac{Re\times \mathit{P}\mathit{r}\times f/2}{1.07+\left({\displaystyle \frac{900}{Re}}\right)-\left(0.63/\left(1+10\times Pr\right)\right)+12.7\times {\left(f/2\right)}^{0.5}\left({Pr}^{0.67}-1\right)}\right){\displaystyle \frac{k}{d_{\mathrm{i}}}},Re>{10}^4,\\ & f=0.00128+0.1143{\left(Re\right)}^{-0.311}.\end{array}

By describing the hydraulic and equivalent diameter instead of the tube diameter, the simulation on the shell side is the same as that on the tube side. The equivalent and hydraulic diameters (D_{\mathrm{e}} and D_{\mathrm{h}}) are employed for computing the Reynolds and Nusselt numbers, as

(9)D_{\mathrm{h}}=D_{\mathrm{s}}-d_{\mathrm{o}},

(10)D_{\mathrm{e}}=\frac{{D_{\mathrm{s}}}^2-{d_{\mathrm{o}}}^2}{d_{\mathrm{o}}},

(11)Nu=h{D}_{\mathrm{e}}/k,

(12)Re=\frac{\rho v{D}_{\mathrm{h}}}{\mu },

where v is the fluid velocity and D_{\mathrm{s}} is the diameter of the shell in terms of the tube number. In addition, P_{\mathrm{t}}\mathrm{a}\mathrm{n}\mathrm{d}CL are the constants of tube pitch and tube arrangement, respectively. Furthermore, CTP is the constant for the tube pass number, which is defined as

(13)D_{\text{s}}=P_{\text{t}}\sqrt{4N_{\text{t}}\times CL\times (1/\text{π}\times CTP)}.

2.2 Fin thermal analysis

The influence of the fins on the thermal efficiency was determined by a parameter called the fin set efficiency; therefore, the efficiency of the fin in the constructal arrangement configurations can be defined as [16]

(14){\eta }_{\mathrm{o}}=1-\left(\left(1-{\eta }_1\right)\frac{A_1}{A}+\left(1-{\eta }_2\right)\frac{A_2}{A}\right),

(15)A=A_{\mathrm{o}}+A_1+A_2,

(16)A_1=N_{\text{t}}\times N_{\text{F}}\times N_{\text{p}}\times l\times (2l_1+t_1),

(17)A_2=2{\times N}_{\text{t}}\times N_{\text{F}}\times N_{\text{p}}\times l\times (2l_2+t_2),

where A_1, A_2, and N_{\text{F}} are heat transfer areas in the first part, the second part, and the total number of fins in the fin array, respectively. Furthermore, l_1, l_2, t_1,\mathrm{a}\mathrm{n}\mathrm{d}{t}_2 are the fin lengths of the first part and second, and fin thicknesses in the first and second parts, respectively. The fin effectiveness, which is obtained from Eq. (18), and the constructal arrangement is estimated as [15]

(18)\eta =Q/Q_{\mathrm{m}\mathrm{a}\mathrm{x}},

(19){\eta }_1=\frac{Q_{\mathrm{F},1}}{Q_{\mathrm{m}\mathrm{a}\mathrm{x},1}}=\frac{Q_{\mathrm{F},1}}{h_s{A}_1{\theta }_{\mathrm{b}}},

(20){\eta }_2=\frac{Q_{\mathrm{F},2}}{Q_{\mathrm{m}\mathrm{a}\mathrm{x},2}}=\frac{Q_{\mathrm{F},2}}{h_{\mathrm{s}}{A}_2{\theta }_{\mathrm{b}}},

(21){\theta }_{\mathrm{b}}=T_{\mathrm{b}}-T_{\mathrm{\infty }},

where Q_{\mathrm{F}}, T_{\mathrm{b}}, and T_{\mathrm{\infty }} are the fin heat transfer, temperature of the base surface, and average temperature of the hot fluid, respectively. Moreover, Q_{\mathrm{F},1} and Q_{\mathrm{F},2} are calculated in the same way as stated in Ref. [15].

In addition, {\mathrm{\Delta }P}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} is the total pressure drop, which is the sum of the tube and shell side pressure drop, which is obtained as [2]

(22){\mathrm{\Delta }P}_{\text{t}}=\frac{{\dot{m}}_t{N}_{\text{p}}}{2{\rho }_{\mathrm{t}}{A_{\text{t}}}^2}\left(\frac{4f_{\text{t}}l}{d_i}+\left(1-{\sigma }_{\mathrm{t}}^2+K_{\text{c,t}}\right)-\left(1-{\sigma }_{\mathrm{t}}^2-K_{\text{e,t}}\right)\right),

(23){\mathrm{\Delta }P}_{\text{s}}=\frac{{\dot{m}}_s{N}_{\text{p}}}{2{\rho }_{\mathrm{s}}{A_{\text{c}}}^2}\left(\frac{4f_{s}l}{D_h}+\left(1-{\sigma }_{\mathrm{s}}^2+K_{\text{c,s}}\right)-\left(1-{\sigma }_{\mathrm{s}}^2-K_{\text{e,s}}\right)\right),

where K_{\text{e}}, \sigma, and K_{\text{c}} are the exit pressure loss coefficient, ratio of minimum free flow area to frontal area, and coefficient of entry pressure loss, respectively. Additionally, A_{\text{t}} and A_{\text{c}} denote the cross-sectional areas on the tube and shell sides, respectively.

3 Objective functions, design parameters, and constraints

3.1 Objective functions

The development of thermal performance by increasing the level of heat transfer has always been considered. Although this increase is accompanied by an improvement in thermal performance (optimal points), on the other hand, it increases the required materials and causes the cost to increase (undesirable amounts). In fact, it can be said that there is a conflict between effectiveness and total heat transfer area, and it is extremely important to find the lowest total heat transfer area with an equal effectiveness. In addition, the use of expanded surfaces with the aim of developing the rate of heat transfer adds to the above problems and doubles the importance of these problems. To investigate the above issues together, multi-objective optimization was performed by simultaneously selecting the effectiveness and total heat transfer area as objective functions. Effectiveness is defined in Eq. (24), and the total heat transfer area of the heat exchanger is estimated as

(24)\epsilon =\frac{\left(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{N}\mathrm{T}\mathrm{U}\left(1-C^{\ast }\right)\right)\right)}{\left(1-C^{\ast }\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{N}\mathrm{T}\mathrm{U}\left(1-C^{\ast }\right)\right)\right)},

where the heat capacity ratio (C^{\ast }) is defined as

(25)C^{\ast }=\frac{C_{\mathrm{m}\mathrm{i}\mathrm{n}}}{C_{\mathrm{m}\mathrm{a}\mathrm{x}}},

(26)C_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\left(\dot{m}{c}_{\mathrm{p}}\right)}_{\mathrm{c}},{\left(\dot{m}{c}_{\mathrm{p}}\right)}_{\mathrm{h}}\right),

(27){\text{C}}_{\text{max}}=\mathrm{m}\mathrm{a}\mathrm{x}\left({\left(\dot{m}{c}_{\mathrm{p}}\right)}_{\mathrm{c}},{\left(\dot{m}{c}_{\mathrm{p}}\right)}_{\mathrm{h}}\right),

(28)A_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=A{\times N}_{\mathrm{M}\mathrm{T}\mathrm{H}\mathrm{E}},

where N_MTHE is number of multi tube heat exchanger.

3.2 Design parameters and constraints

Two types of design parameters, including the fin-related parameters and heat exchanger parameters, were considered. The tubular heat exchanger network geometry, including the tube diameter (inner and outer), tube length, tube number, tube arrangement, tube pitch ratio, number of heat exchangers, and constructal fin geometry, including fin length in the first part, fin thickness in the first part, fin length in the second part, fin thickness in the second part, and fin number (as illustrated in Fig.1 and Fig.2), were selected as 11 decision variables. It should be noted that the width of the gap between the two branches of the second part of the fins does not affect the thermal performance of the system. In fact, because one-dimensional heat transfer is considered, the value of the mentioned gap does not affect the rate of heat transfer and, consequently, should not be considered as an additional design parameter. To obtain an accurate and appropriate optimization program, the relevant constraints should be considered in the optimization program. The pressure drop is determined to be less than 100 kPa on each side.

(29)

3.3 Case study

In this study, different types of fins, including simple fins and constructal fins, were chosen and applied in the THEN, and the results are compared with those of a conventional heat exchanger without fins. The oil with a mass flow rate of 1 kg/s and an inlet temperature of 80 °C was selected as the hot fluid, while fresh water with a mass flow rate of 3 kg/s and an inlet temperature of 20 °C was selected as the cold fluid. The variation of thermophysical properties with temperature was applied, and the required properties were determined at the average inlet and outlet temperatures of each HE. The decision variables and range of their changes are presented in Tab.1. Thirty-eight standard tubes with diameters listed in Tab.2 are deliberated for the selection of the tube diameter [2]. It should be noted that the mentioned tube diameters should be considered as discrete design parameters. One of the advantages of genetic algorithms is the handling of discrete parameters. For this purpose, the available diameters were provided in a matrix. Then, a random number between 1 and 38 (total available tube diameter) was generated and rounded. After that, based on this number, the tube diameter was selected.

3.4 Model validation

To validate the numerical simulation results, the numerical results were evaluated using the experimental data provided by Kakac et al. [1]. It is worth mentioning that the validation is related to a simple tube case (carbon steel), which is shown in Fig.3(a), and the results of this comparison are shown in Fig.3(b). Furthermore, in the simple fin case, the comparison of the present modeling results and the corresponding values from Kakac et al. [1] for the same input values is summarized in Tab.3. The experimental model contained an exchanger with 28 tubes with inner and outer tube diameters of 10 and 12 mm, respectively. The length of the tube was 0.6 m and the inner diameter of the shell side was 0.11 m. Water with a mass flow rate of 1 L/min and an inlet temperature of 60 °C were considered on the shell side, while water with an inlet temperature of 22 °C is on the tube side. Because of the short length of the tubes and elimination of the entrance effects in the numerical model, the thermal efficiency was slightly higher in the experimental model. The experimental model features are presented in Tab.4. Fig.3(b) and Tab.3 present the validation results. The numerical and experimental data are in good agreement with the maximum error bar of 13.3 percent.

4 Results and discussion

The NSGA-II was applied for 1000 generations, using a search population size of 150 individuals, gene mutation probability of 0.035, crossover probability of 0.9, and controlled elitism value of 0.5. The results for the Pareto optimal front are depicted in Fig.4. To simplify the results, the optimization was performed with and without extended surfaces (fins). The results in Fig.4 show that the best objective function results are obtained by adding fins compared to those without fins. On the other hand, the optimum Pareto front for the case with a constructal fin is completely dominated by the optimum results in the case of a simple fin and without a fin. Furthermore, for the desirable value of effectiveness, the constructal fin needs a lower total heat transfer area compared with the simple fin and without fin (usual heat exchanger).

To conclude the optimization problem, it is necessary to familiarize an optimum point among the optimal points of the Pareto front. For this purpose, the LINMAP procedure was used [14]. The final values of the two objective functions for a case without fins as well as with fins (simple fin and constructal fin) are presented in Tab.5 for the Pareto front in Fig.4, and the improvement of each of the objective functions in the simple fin and constructal fin cases are compared to those without fins, and the results are presented in Tab.6. Another important parameter is the pressure drop, which is also calculated and listed in Tab.7 for the final optimum solution. As shown, the total pressure drop (sum of cold and hot sides) for the three studied cases are relatively in the same range. In fact, the optimization algorithm changes the parameters for the constructal tubes to moderate the pressure drop. For example, as shown in Tab.7, the inner diameter of the tube for the constructal fin is selected to be higher than that of the simple fin. In addition, this parameter is higher in the case of a simple fin than in a plain tube. Another important parameter that moderates the pressure drop on the fin side is the tube pitch ratio, which is selected to be higher in the case of constructal than in the simple fin.

Fig.4 shows the addition of a simple fin to the inner tube of the heat exchanger enhances the effectiveness from 0.5613 up to 0.5911 while using constructal fins shows a 7% increment of effectiveness (0.6007). Also, the other objective function (total heat transfer area) in simple fin underwent a 4.0656% increase that is not very suitable for building a heat exchanger due to the manufacturing limitations and material properties. However, this parameter is significantly reduced by about 9.4763% in constructal fin compared to the usual heat exchanger, decreasing the heat transfer area required in exchangers and consequently, reducing the required materials and cost.

Using fins expands the optimization and increases the heat transfer between the cold and hot fluids by enhancing the effective contact area of the fins. The respective thermal surfaces provided by the fins and tubes are shown separately in Fig.5(a) and Fig.5(b). Furthermore, the area of heat transfer is indicated by the fin. Even in the constructal fin case, the importance of the fin is increased to its highest optimal point and the fin surfaces contribute more than the tube surfaces to the heat transfer process and optimization of the system (Fig.5(c) and Fig.5(d)).

As shown in Fig.4, achieving a higher thermal efficiency in a constant total heat transfer area is possible using the fins. This confirms the enhancement of the heat transfer rate in a constant total heat transfer area. Improving the thermal parameters will enhance the heat transfer rate between cold and hot fluids. Furthermore, the heat transfer rate is enhanced because of the higher effectiveness of the cold and hot fluids in the heat exchanger. As a result, the amount of the heat transferred from the hot fluid to the cold fluid increases, and the temperature of the cold and hot fluids increases or decreases in larger values. This behavior is shown in Fig.6. In addition, the growth of the heat transfer rate, maximum temperature of water, and minimum temperature of oil in the constructal fin case, which has been reported in Fig.6, occurs because of the maximum efficiency of the heat exchanger.

In Fig.7(a), it can be seen that the number of tubes, especially in the constructal fin case, decreases significantly, which can help to achieve an optimum design with a lower number of tubes and reduce the expenses. Moreover, reducing the tube length effects on the constructal fin case can be seen in Fig.7(b). The distribution of the number of heat exchangers versus effectiveness is shown in Fig.7(c), which confirms the improvement in thermal performance after using the fin and its increase in the constructal fin design. It can be observed in Fig.7(c) that by investigating each specific number of heat exchangers with two different fin shapes, a larger distribution of effectiveness can be achieved and distribution increases in maximum effectiveness with a specific number of heat exchangers.

Reducing the number of tubes in the fins (Fig.7(a)) increases the inlet velocity and consequently increases the Reynolds number on the tube side, while increasing the Nusselt number affects the pressure drop and increases the hydraulic parameter. On the heat exchanger shell side, to reduce the number of tubes and consequently reduce the shell diameter, the use of the fins also plays a role in reducing the cross-sectional area of the shell fluid and increasing the Reynolds number, consequently increasing the pressure drop. Compensation for the increase in pressure drop for both sides of the tube and the shell in the optimal position is achieved by two design parameters, tube pitch ratio and tube diameter, respectively, as shown in Fig.8(a) and Fig.8(b), the inner diameter increases on the tube side, and higher values are considered. The highest growth rate occurred in the mode constructal fin, while for the other side, the tube pitch ratio increased in order to increase the cross-sectional area of the shell fluid, and the result is to compensate for the increase in pressure drop due to the use of the fin.

The volume of the heat exchanger at the optimal points presented in Fig.4 is calculated, and its distribution versus effectiveness is shown in Fig.9 where there is a clear conflict between effectiveness and this characteristic of the heat exchanger (volume MTHE), which shows that increasing the effectiveness of the heat exchanger is accompanied by an increase in its volume, and the heat exchanger with a higher effectiveness is more voluminous. In the optimal position, using the fins, on the one hand, a smaller number of tubes is needed and the tube length is reduced; hence, the tube pitch ratio and tube inner diameter show a slight increase, and the decrease in the number of tubes and tube length overcomes the enhancement in the inner diameter of the tube and tube pitch ratio, and eventually the volume of the heat exchanger decreases. In addition, in the case of the constructal fin, according to Fig.9, owing to the enhancement in differences, the lowest volume of the exchanger occurs, and the exchanger can be used as an effective method to reduce the volume of the constructal fin, especially at high capacities.

5 Conclusions

In the present work, a multi-objective optimization design procedure based on the constructal theory is suggested for a THEN which was optimally designed by defining two objective functions (effectiveness and total heat transfer area) using a genetic algorithm procedure. The effectiveness was maximized, and the total heat transfer area was minimized. The contributions of the obtained results are as follows.

Adding fins to the inner tube improves the optimized objective functions in a way that allows achieving a higher thermal effectiveness at a constant total heat transfer area, especially for the constructal fin case.

The effect of the fin on the thermal parameters proves that the heat transfer rate increases at a constant total heat transfer area, which leads to an increase in the temperature in the cold fluid and a reduction in the temperature of the hot fluid. The highest temperature difference between the hot and cold fluids was achieved by the constructal fin case.

Using the fin and providing the essential thermal surface for the heat exchanger at the optimum position reduces the effect of the tube and heat transfer area in the tubes.

Heat exchangers equipped with fins, especially at high values of effectiveness, need to have a lower tube number and smaller tube length in such a way that using a constructal fin minimizes the number of necessary tubes and their tube length.

Although the tube diameter and tube pitch ratio increases with the addition of fins, owing to the reduction in the number and length of tubes, especially at higher values of effectiveness, the volume of the heat exchanger decreases significantly.

6 Notations

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